Conjectured Explicit Formulas Enumerating 5-rowed Truncated Standard Young \

    Tableau of shape [n,n+a1,n+a2,n+a3,n+a4] for a from 0<=a1,a2,a3,a4<= , 5,

    and a4<a3, divided by the numer of SYT of shape , [n, n, n, n, n]



                              By Shalosh B. Ekhad





Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                   5 (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
                   -----------------------------------------
                   8 (4 n + 1) (5 + 4 n) (2 n + 1) (4 n + 3)



                             and in Maple notation



5/8*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(4*n+1)/(5+4*n)/(2*n+1)/(4*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              15 (3 n + 2) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
              ----------------------------------------------------
              16 (4 n + 1) (5 + 4 n) (2 n + 1) (2 n + 3) (4 n + 3)



                             and in Maple notation



15/16*(3*n+2)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(4*n+1)/(5+4*n)/(2*n+1)/(2*n+3)/(
4*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             5 (13 n + 12) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
             -----------------------------------------------------
             16 (4 n + 1) (5 + 4 n) (2 n + 1) (2 n + 3) (4 n + 3)



                             and in Maple notation



5/16*(13*n+12)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(4*n+1)/(5+4*n)/(2*n+1)/(2*n+3)/
(4*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                2
         5 (71 n  + 107 n + 42) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
         --------------------------------------------------------------
         16 (4 n + 1) (5 + 4 n) (2 n + 1) (2 n + 3) (4 n + 3) (4 n + 7)



                             and in Maple notation



5/16*(71*n^2+107*n+42)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(4*n+1)/(5+4*n)/(2*n+1)/
(2*n+3)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                 2
         15 (41 n  + 82 n + 42) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
         --------------------------------------------------------------
         16 (4 n + 1) (5 + 4 n) (2 n + 1) (2 n + 3) (4 n + 3) (4 n + 7)



                             and in Maple notation



15/16*(41*n^2+82*n+42)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(4*n+1)/(5+4*n)/(2*n+1)/
(2*n+3)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



         175 (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)
         --------------------------------------------------------------
         16 (4 n + 1) (5 + 4 n) (2 n + 1) (2 n + 3) (4 n + 3) (4 n + 7)



                             and in Maple notation



175/16*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(4*n+1)/(5+4*n)/(2*n+1)/(2
*n+3)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                 3        2
15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (173 n  + 437 n  + 378 n + 112)/(64

    (n + 2) (4 n + 7) (4 n + 3) (2 n + 3) (2 n + 1) (5 + 4 n) (4 n + 1))



                             and in Maple notation



15/64*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(173*n^3+437*n^2+378*n+112)/(n+2)/(4*n+7)
/(4*n+3)/(2*n+3)/(2*n+1)/(5+4*n)/(4*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                 3         2
15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (337 n  + 1093 n  + 1202 n + 448)/(

    64 (n + 2) (4 n + 7) (4 n + 3) (2 n + 3) (2 n + 1) (5 + 4 n) (4 n + 1))



                             and in Maple notation



15/64*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(337*n^3+1093*n^2+1202*n+448)/(n+2)/(4*n+
7)/(4*n+3)/(2*n+3)/(2*n+1)/(5+4*n)/(4*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                 3         2
5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (1711 n  + 6219 n  + 7526 n + 3024)/(

    64 (n + 2) (4 n + 7) (4 n + 3) (2 n + 3) (2 n + 1) (5 + 4 n) (4 n + 1))



                             and in Maple notation



5/64*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(1711*n^3+6219*n^2+7526*n+3024)/(n+2)/(4*n
+7)/(4*n+3)/(2*n+3)/(2*n+1)/(5+4*n)/(4*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                 3         2
5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (2411 n  + 9159 n  + 11446 n + 4704)/

    (64 (n + 2) (4 n + 7) (4 n + 3) (2 n + 3) (2 n + 1) (5 + 4 n) (4 n + 1))



                             and in Maple notation



5/64*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(2411*n^3+9159*n^2+11446*n+4704)/(n+2)/(4*
n+7)/(4*n+3)/(2*n+3)/(2*n+1)/(5+4*n)/(4*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
15 (601 n  + 1639 n  + 1546 n + 504) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (1 + n)/(32 (2 n + 1) (2 n + 3) (n + 2) (4 n + 3) (4 n + 7) (4 n + 1)

    (5 + 4 n) (4 n + 9))



                             and in Maple notation



15/32*(601*n^3+1639*n^2+1546*n+504)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(2*n+
1)/(2*n+3)/(n+2)/(4*n+3)/(4*n+7)/(4*n+1)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                  2
75 (6 n + 7) (17 n  + 39 n + 24) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 6)/(64 (2 n + 1) (2 n + 3) (n + 2) (4 n + 3) (4 n + 7) (4 n + 1)

    (5 + 4 n) (4 n + 9))



                             and in Maple notation



75/64*(6*n+7)*(17*n^2+39*n+24)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)/(2*n+1)/
(2*n+3)/(n+2)/(4*n+3)/(4*n+7)/(4*n+1)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1)

           4          3          2
    (7247 n  + 37965 n  + 74560 n  + 64980 n + 21168)/(32 (2 n + 1) (2 n + 3)

    (n + 2) (4 n + 3) (4 n + 7) (4 n + 1) (5 + 4 n) (4 n + 9))



                             and in Maple notation



5/32*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(7247*n^4+37965*n^3+74560*n^2+64980*n+
21168)/(2*n+1)/(2*n+3)/(n+2)/(4*n+3)/(4*n+7)/(4*n+1)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1)

           4          3          2
    (8046 n  + 44755 n  + 92445 n  + 83930 n + 28224)/(64 (2 n + 1) (2 n + 3)

    (n + 2) (4 n + 3) (4 n + 7) (4 n + 1) (5 + 4 n) (4 n + 9))



                             and in Maple notation



15/64*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(8046*n^4+44755*n^3+92445*n^2+83930*n+
28224)/(2*n+1)/(2*n+3)/(n+2)/(4*n+3)/(4*n+7)/(4*n+1)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1)

            4          3           2
    (16891 n  + 96300 n  + 202775 n  + 186810 n + 63504)/(32 (2 n + 1)

    (2 n + 3) (n + 2) (4 n + 3) (4 n + 7) (4 n + 1) (5 + 4 n) (4 n + 9))



                             and in Maple notation



5/32*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(16891*n^4+96300*n^3+202775*n^2+186810*n+
63504)/(2*n+1)/(2*n+3)/(n+2)/(4*n+3)/(4*n+7)/(4*n+1)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                   2
              5 (5 n + 3) (5 n + 2) (5 n + 1) (61 n  + 155 n + 84)
              ----------------------------------------------------
              9 (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 5) (3 n + 2)



                             and in Maple notation



5/9*(5*n+3)*(5*n+2)*(5*n+1)*(61*n^2+155*n+84)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+5)/(
3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                   2
              5 (5 n + 3) (5 n + 2) (5 n + 1) (43 n  + 69 n + 28)
              ---------------------------------------------------
              3 (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 5) (3 n + 2)



                             and in Maple notation



5/3*(5*n+3)*(5*n+2)*(5*n+1)*(43*n^2+69*n+28)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+5)/(3
*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             10 (19 n + 21) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
             ------------------------------------------------------
              9 (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 5) (3 n + 2)



                             and in Maple notation



10/9*(19*n+21)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+5)/
(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        2
         5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (49 n  + 89 n + 42)
         -------------------------------------------------------------
          9 (n + 2) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 5) (3 n + 2)



                             and in Maple notation



5/9*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(49*n^2+89*n+42)/(n+2)/(7+3*n)/(4+3*n)/(1+3
*n)/(3*n+5)/(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        2
         5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (29 n  + 69 n + 42)
         -------------------------------------------------------------
          3 (n + 2) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 5) (3 n + 2)



                             and in Maple notation



5/3*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(29*n^2+69*n+42)/(n+2)/(7+3*n)/(4+3*n)/(1+3
*n)/(3*n+5)/(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



         25 (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)
         --------------------------------------------------------------
          9 (n + 2) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 5) (3 n + 2)



                             and in Maple notation



25/9*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(n+2)/(7+3*n)/(4+3*n)/(1+3
*n)/(3*n+5)/(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          5           4            3            2
5 (89287 n  + 581926 n  + 1504219 n  + 1932276 n  + 1237572 n + 317520)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)/(72 (3 n + 2) (3 n + 5)

    (2 n + 3) (n + 2) (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n) (7 + 3 n)

    (4 n + 7))



                             and in Maple notation



5/72*(89287*n^5+581926*n^4+1504219*n^3+1932276*n^2+1237572*n+317520)*(5*n+1)*(5
*n+2)*(5*n+3)*(5*n+4)*(1+n)/(3*n+2)/(3*n+5)/(2*n+3)/(n+2)/(5+4*n)/(4*n+9)/(1+3*
n)/(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4         3          2
25 (1427 n  + 8166 n  + 17521 n  + 16758 n + 6048) (5 n + 1) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)/(72 (3 n + 2) (3 n + 5) (2 n + 3)

    (n + 2) (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/72*(1427*n^4+8166*n^3+17521*n^2+16758*n+6048)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4
)*(5*n+6)*(5*n+7)/(3*n+2)/(3*n+5)/(2*n+3)/(n+2)/(5+4*n)/(4*n+9)/(1+3*n)/(4+3*n)
/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



         4         3          2
25 (817 n  + 5026 n  + 11579 n  + 11842 n + 4536) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 6) (5 n + 7)/(24 (3 n + 2) (3 n + 5) (2 n + 3) (n + 2)

    (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/24*(817*n^4+5026*n^3+11579*n^2+11842*n+4536)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)
*(5*n+6)*(5*n+7)/(3*n+2)/(3*n+5)/(2*n+3)/(n+2)/(5+4*n)/(4*n+9)/(1+3*n)/(4+3*n)/
(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



         2
25 (139 n  + 407 n + 294) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6)

    (5 n + 7) (5 n + 8) (5 n + 9)/(72 (3 n + 2) (3 n + 5) (2 n + 3) (n + 2)

    (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/72*(139*n^2+407*n+294)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)*(5*n+
8)*(5*n+9)/(3*n+2)/(3*n+5)/(2*n+3)/(n+2)/(5+4*n)/(4*n+9)/(1+3*n)/(4+3*n)/(7+3*n
)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4         3          2
25 (1621 n  + 8602 n  + 17029 n  + 15028 n + 5040) (5 n + 1) (5 n + 2)

    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(48 (3 n + 2) (3 n + 5)

    (2 n + 3) (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/48*(1621*n^4+8602*n^3+17029*n^2+15028*n+5040)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4
)*(1+n)*(5*n+6)*(5*n+7)/(3*n+2)/(3*n+5)/(2*n+3)/(5+2*n)/(n+2)/(5+4*n)/(4*n+9)/(
1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5          4           3           2
25 (10571 n  + 77603 n  + 228637 n  + 338413 n  + 251976 n + 75600) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)/(144 (3 n + 2) (3 n + 5)

    (2 n + 3) (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/144*(10571*n^5+77603*n^4+228637*n^3+338413*n^2+251976*n+75600)*(5*n+1)*(5*n+
2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(3*n+2)/(3*n+5)/(2*n+3)/(5+2*n)/(n+2)/(5+4*n
)/(4*n+9)/(1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            2       2
25 (5 n + 7)  (163 n  + 517 n + 420) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 6) (5 n + 8) (5 n + 9)/(144 (3 n + 2) (3 n + 5) (2 n + 3) (5 + 2 n)

    (n + 2) (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/144*(5*n+7)^2*(163*n^2+517*n+420)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5
*n+8)*(5*n+9)/(3*n+2)/(3*n+5)/(2*n+3)/(5+2*n)/(n+2)/(5+4*n)/(4*n+9)/(1+3*n)/(4+
3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
25 (457 n  + 2248 n  + 3655 n + 1960) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)/(48 (3 n + 2) (3 n + 5) (2 n + 3)

    (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n) (7 + 3 n)

    (4 n + 7))



                             and in Maple notation



25/48*(457*n^3+2248*n^2+3655*n+1960)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5
*n+7)*(5*n+8)*(5*n+9)/(3*n+2)/(3*n+5)/(2*n+3)/(5+2*n)/(n+2)/(5+4*n)/(4*n+9)/(1+
3*n)/(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, 1 + n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



          3         2
25 (1927 n  + 9762 n  + 16211 n + 8820) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)/(144 (3 n + 2) (3 n + 5) (2 n + 3)

    (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9) (1 + 3 n) (4 + 3 n) (7 + 3 n)

    (4 n + 7))



                             and in Maple notation



25/144*(1927*n^3+9762*n^2+16211*n+8820)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)
*(5*n+7)*(5*n+8)*(5*n+9)/(3*n+2)/(3*n+5)/(2*n+3)/(5+2*n)/(n+2)/(5+4*n)/(4*n+9)/
(1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                               3        2
        40 (5 n + 3) (5 n + 2) (5 n + 1) (139 n  + 496 n  + 517 n + 168)
        ----------------------------------------------------------------
         9 (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2)



                             and in Maple notation



40/9*(5*n+3)*(5*n+2)*(5*n+1)*(139*n^3+496*n^2+517*n+168)/(7+3*n)/(4+3*n)/(1+3*n
)/(3*n+8)/(3*n+5)/(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                2
        35 (63 n  + 109 n + 48) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
        ---------------------------------------------------------------
         9 (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2)



                             and in Maple notation



35/9*(63*n^2+109*n+48)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(7+3*n)/(4+3*n)/(1+3*n)/
(3*n+8)/(3*n+5)/(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         700 (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)
         -------------------------------------------------------------
         9 (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2)



                             and in Maple notation



700/9*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(7+3*n)/(4+3*n)/(1+3*n)/(3*
n+8)/(3*n+5)/(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           4          3           2
25 (16483 n  + 85846 n  + 164323 n  + 137172 n + 42336) (5 n + 1) (5 n + 2)

    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)/(36 (2 n + 3) (5 + 4 n) (4 n + 9)

    (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/36*(16483*n^4+85846*n^3+164323*n^2+137172*n+42336)*(5*n+1)*(5*n+2)*(5*n+3)*(
5*n+4)*(1+n)*(5*n+6)/(2*n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4
+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           4           3           2
25 (30819 n  + 182614 n  + 405347 n  + 399924 n + 148176) (5 n + 1) (5 n + 2)

    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)/(36 (2 n + 3) (5 + 4 n) (4 n + 9)

    (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/36*(30819*n^4+182614*n^3+405347*n^2+399924*n+148176)*(5*n+1)*(5*n+2)*(5*n+3)
*(5*n+4)*(1+n)*(5*n+6)/(2*n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/
(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          3          2
25 (9031 n  + 43233 n  + 68748 n + 36288) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(36 (2 n + 3) (5 + 4 n) (4 n + 9)

    (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/36*(9031*n^3+43233*n^2+68748*n+36288)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*
(5*n+6)*(5*n+7)/(2*n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)
/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6           5            4            3            2
25 (114079 n  + 979459 n  + 3466134 n  + 6476183 n  + 6744257 n  + 3715488 n

     + 846720) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)/(72

    (2 n + 3) (5 + 2 n) (n + 2) (4 n + 7) (3 n + 2) (3 n + 5) (3 n + 8)

    (1 + 3 n) (4 + 3 n) (7 + 3 n) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/72*(114079*n^6+979459*n^5+3466134*n^4+6476183*n^3+6744257*n^2+3715488*n+
846720)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)/(4*
n+7)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5           4            3            2
25 (47471 n  + 386472 n  + 1257130 n  + 2044899 n  + 1665648 n + 544320)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(72

    (2 n + 3) (5 + 2 n) (n + 2) (4 n + 7) (3 n + 2) (3 n + 5) (3 n + 8)

    (1 + 3 n) (4 + 3 n) (7 + 3 n) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/72*(47471*n^5+386472*n^4+1257130*n^3+2044899*n^2+1665648*n+544320)*(5*n+1)*(
5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+3)/(5+2*n)/(n+2)/(4*n+7)/(3*n
+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4          3           2
25 (5573 n  + 39214 n  + 103320 n  + 120833 n + 52920) (5 n + 1) (5 n + 2)

    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)/(24 (2 n + 3)

    (5 + 2 n) (n + 2) (4 n + 7) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n)

    (4 + 3 n) (7 + 3 n) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/24*(5573*n^4+39214*n^3+103320*n^2+120833*n+52920)*(5*n+1)*(5*n+2)*(5*n+3)*(5
*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+3)/(5+2*n)/(n+2)/(4*n+7)/(3*n+2)/(3*n+
5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



            5            4            3            2
25 (119719 n  + 1057452 n  + 3724742 n  + 6539451 n  + 5721936 n + 1995840)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(72

    (2 n + 3) (5 + 2 n) (n + 2) (4 n + 7) (3 n + 2) (3 n + 5) (3 n + 8)

    (1 + 3 n) (4 + 3 n) (7 + 3 n) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/72*(119719*n^5+1057452*n^4+3724742*n^3+6539451*n^2+5721936*n+1995840)*(5*n+1
)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+3)/(5+2*n)/(n+2)/(4*n+7)/(
3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5            4            3            2
25 (152061 n  + 1331693 n  + 4821013 n  + 9244375 n  + 9912554 n  + 5640144 n

     + 1330560) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)

    (5 n + 7)/(72 (2 n + 3) (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9) (3 n + 2)

    (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/72*(152061*n^6+1331693*n^5+4821013*n^4+9244375*n^3+9912554*n^2+5640144*n+
1330560)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+3)/(5+2*n)/
(n+2)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)/(
4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5           4            3            2
25 (68389 n  + 570530 n  + 1907297 n  + 3198805 n  + 2695719 n + 914760)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)

    (5 n + 8)/(72 (2 n + 3) (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9) (3 n + 2)

    (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/72*(68389*n^5+570530*n^4+1907297*n^3+3198805*n^2+2695719*n+914760)*(5*n+1)*(
5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+3)/(5+2*n)/(n+2)/(5+4
*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                4           3           2
    (5 n + 2) (5 n + 1) (27053 n  + 196306 n  + 534889 n  + 648872 n + 295680)/

    (72 (2 n + 3) (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5)

    (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/72*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
27053*n^4+196306*n^3+534889*n^2+648872*n+295680)/(2*n+3)/(5+2*n)/(n+2)/(5+4*n)/
(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             6             5             4              3              2
25 (1155201 n  + 12753872 n  + 58536991 n  + 142951363 n  + 195879833 n

     + 142776240 n + 43243200) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)

    (5 n + 6) (5 n + 7)/(72 (2 n + 3) (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9)

    (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



25/72*(1155201*n^6+12753872*n^5+58536991*n^4+142951363*n^3+195879833*n^2+
142776240*n+43243200)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*
n+3)/(5+2*n)/(n+2)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3
*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 2, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



             6             5             4              3              2
25 (1634077 n  + 18300589 n  + 85067931 n  + 210081329 n  + 290701538 n

     + 213700896 n + 65197440) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)

    (5 n + 6) (5 n + 7)/(72 (2 n + 3) (5 + 2 n) (n + 2) (5 + 4 n) (4 n + 9)

    (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



25/72*(1634077*n^6+18300589*n^5+85067931*n^4+210081329*n^3+290701538*n^2+
213700896*n+65197440)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*
n+3)/(5+2*n)/(n+2)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3
*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



10 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1)

           4         3          2
    (1387 n  + 8232 n  + 17213 n  + 14832 n + 4536)/(27 (n + 3) (n + 2)

    (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2))



                             and in Maple notation



10/27*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(1387*n^4+8232*n^3+17213*n^2+14832*n+4536
)/(n+3)/(n+2)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+8)/(3*n+5)/(3*n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7             6              5              4              3
5 (2621867 n  + 28364119 n  + 129149889 n  + 320834665 n  + 469720328 n

                  2
     + 405495516 n  + 191295216 n + 38102400) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n)/(72 (2 n + 3) (n + 2) (n + 3) (5 + 4 n) (4 n + 9)

    (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



5/72*(2621867*n^7+28364119*n^6+129149889*n^5+320834665*n^4+469720328*n^3+
405495516*n^2+191295216*n+38102400)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(2*n+
3)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/
(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5             4              3
25 (546089 n  + 6173688 n  + 29541560 n  + 77584782 n  + 120824351 n

                  2
     + 111616842 n  + 56654640 n + 12192768) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 1) (5 n + 6)/(216 (2 n + 3) (n + 2) (n + 3) (5 + 4 n) (4 n + 9)

    (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/216*(546089*n^7+6173688*n^6+29541560*n^5+77584782*n^4+120824351*n^3+
111616842*n^2+56654640*n+12192768)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)/(2*n
+3)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)
/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4              3
25 (2214349 n  + 24288216 n  + 112451992 n  + 284986950 n  + 427209691 n

                  2
     + 379111794 n  + 184606128 n + 38102400) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6)/(432 (2 n + 3) (5 + 2 n) (n + 2) (n + 3)

    (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/432*(2214349*n^7+24288216*n^6+112451992*n^5+284986950*n^4+427209691*n^3+
379111794*n^2+184606128*n+38102400)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/
(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5             4             3
25 (293247 n  + 3366668 n  + 16396220 n  + 43924870 n  + 69927973 n

                 2
     + 66169662 n  + 34462800 n + 7620480) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 1) (5 n + 7) (5 n + 6)/(144 (2 n + 3) (5 + 2 n) (n + 2) (n + 3)

    (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/144*(293247*n^7+3366668*n^6+16396220*n^5+43924870*n^4+69927973*n^3+66169662*
n^2+34462800*n+7620480)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(2*n+3)
/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7
+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 2) (5 n + 3) (5 n + 4)

            5           4            3            2
    (55417 n  + 470485 n  + 1566425 n  + 2553875 n  + 2036838 n + 635040)

    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)/(432 (2 n + 3) (5 + 2 n)

    (n + 2) (n + 3) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n)

    (4 + 3 n) (7 + 3 n) (4 n + 7))



                             and in Maple notation



25/432*(5*n+2)*(5*n+3)*(5*n+4)*(55417*n^5+470485*n^4+1566425*n^3+2553875*n^2+
2036838*n+635040)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)
/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4              3
25 (2985311 n  + 33210582 n  + 156311006 n  + 403745436 n  + 618571451 n

                  2
     + 562632870 n  + 281584944 n + 59875200) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(432 (2 n + 3) (5 + 2 n) (n + 2)

    (n + 3) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n)

    (4 + 3 n) (7 + 3 n) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/432*(2985311*n^7+33210582*n^6+156311006*n^5+403745436*n^4+618571451*n^3+
562632870*n^2+281584944*n+59875200)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/
(1+3*n)/(4+3*n)/(7+3*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5             4             3             2
25 (260171 n  + 2566341 n  + 10420199 n  + 22309467 n  + 26576570 n

     + 16702932 n + 4324320) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9)

    (5 n + 8) (5 n + 7) (5 n + 6)/(432 (2 n + 3) (5 + 2 n) (n + 2) (n + 3)

    (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/432*(260171*n^6+2566341*n^5+10420199*n^4+22309467*n^3+26576570*n^2+16702932*
n+4324320)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n
+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)
/(7+3*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5            4             3             2
25 (160613 n  + 1685956 n  + 7287078 n  + 16585214 n  + 20938849 n

     + 13882770 n + 3769920) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9)

    (5 n + 8) (5 n + 7) (5 n + 6)/(144 (2 n + 3) (5 + 2 n) (n + 2) (n + 3)

    (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/144*(160613*n^6+1685956*n^5+7287078*n^4+16585214*n^3+20938849*n^2+13882770*n
+3769920)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+
3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/
(7+3*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                           6            5            4
175 (5 n + 2) (5 n + 3) (5 n + 4) (100501 n  + 1078485 n  + 4757467 n

                 3             2
     + 11028831 n  + 14150932 n  + 9513384 n + 2613600) (5 n + 1) (5 n + 9)

    (5 n + 8) (5 n + 7) (5 n + 6)/(432 (2 n + 3) (5 + 2 n) (n + 2) (n + 3)

    (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (4 n + 7) (4 n + 11))



                             and in Maple notation



175/432*(5*n+2)*(5*n+3)*(5*n+4)*(100501*n^6+1078485*n^5+4757467*n^4+11028831*n^
3+14150932*n^2+9513384*n+2613600)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+
3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/
(7+3*n)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                 6            5            4             3
    (5 n + 2) (5 n + 1) (259957 n  + 2464255 n  + 9583769 n  + 19610677 n

                 2                           /
     + 22312294 n  + 13405848 n + 3326400)  /  (576 (4 n + 11) (4 n + 7)
                                           /

    (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2) (4 n + 9)

                             2
    (5 + 4 n) (n + 2) (n + 3)  (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/576*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
259957*n^6+2464255*n^5+9583769*n^4+19610677*n^3+22312294*n^2+13405848*n+3326400
)/(4*n+11)/(4*n+7)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+8)/(3*n+5)/(3*n+2)/(4*n+9)/(5+4
*n)/(n+2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4              3
25 (1820555 n  + 21560052 n  + 108620960 n  + 301863594 n  + 499788797 n

                  2
     + 492884994 n  + 267929208 n + 61871040) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (1728 (4 n + 11)
                                                      /

    (4 n + 7) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2)

                                       2
    (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/1728*(1820555*n^7+21560052*n^6+108620960*n^5+301863594*n^4+499788797*n^3+
492884994*n^2+267929208*n+61871040)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*(5*
n+7)*(5*n+8)*(5*n+9)/(4*n+11)/(4*n+7)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+8)/(3*n+5)/(
3*n+2)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4               3
25 (3747911 n  + 47573592 n  + 256760312 n  + 763220970 n  + 1348122689 n

                   2
     + 1413276798 n  + 812947968 n + 197588160) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (1728 (4 n + 11)
                                                      /

    (4 n + 7) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2)

                                       2
    (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/1728*(3747911*n^7+47573592*n^6+256760312*n^5+763220970*n^4+1348122689*n^3+
1413276798*n^2+812947968*n+197588160)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*(
5*n+7)*(5*n+8)*(5*n+9)/(4*n+11)/(4*n+7)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+8)/(3*n+5)
/(3*n+2)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4              3
25 (2187313 n  + 28737752 n  + 160187376 n  + 490551822 n  + 890256863 n

                  2
     + 956109946 n  + 561751008 n + 139043520) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (576 (4 n + 11)
                                                      /

    (4 n + 7) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2)

                                       2
    (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/576*(2187313*n^7+28737752*n^6+160187376*n^5+490551822*n^4+890256863*n^3+
956109946*n^2+561751008*n+139043520)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*(5
*n+7)*(5*n+8)*(5*n+9)/(4*n+11)/(4*n+7)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+8)/(3*n+5)/
(3*n+2)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 3, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7              6              5               4               3
25 (9375967 n  + 124852920 n  + 704363944 n  + 2180089962 n  + 3993418489 n

                   2
     + 4323382878 n  + 2557558080 n + 636672960) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (1728 (4 n + 11)
                                                      /

    (4 n + 7) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2)

                                       2
    (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/1728*(9375967*n^7+124852920*n^6+704363944*n^5+2180089962*n^4+3993418489*n^3+
4323382878*n^2+2557558080*n+636672960)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*
(5*n+7)*(5*n+8)*(5*n+9)/(4*n+11)/(4*n+7)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+8)/(3*n+5
)/(3*n+2)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4          3          2
40 (3609 n  + 22576 n  + 50009 n  + 45966 n + 15120) (5 n + 1) (5 n + 2)

    (5 n + 3) (5 n + 4) (1 + n)/(27 (n + 2) (n + 3) (3 n + 2) (3 n + 5)

    (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



40/27*(3609*n^4+22576*n^3+50009*n^2+45966*n+15120)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n
+4)*(1+n)/(n+2)/(n+3)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (5114157 n  + 69576094 n  + 407474486 n  + 1341473160 n  + 2715321133 n

                   3               2
     + 3461211986 n  + 2714838744 n  + 1199066400 n + 228614400) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)/(144 (2 n + 3) (5 + 2 n)

    (n + 2) (n + 3) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n)

    (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7))



                             and in Maple notation



25/144*(5114157*n^8+69576094*n^7+407474486*n^6+1341473160*n^5+2715321133*n^4+
3461211986*n^3+2714838744*n^2+1199066400*n+228614400)*(5*n+1)*(5*n+2)*(5*n+3)*(
5*n+4)*(1+n)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+5
)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6              5               4
25 (1891421 n  + 26671882 n  + 162583042 n  + 559578840 n  + 1189594689 n

                   3               2
     + 1599898238 n  + 1329740928 n  + 624648960 n + 127008000) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)/(144 (2 n + 3) (5 + 2 n)

    (n + 2) (n + 3) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n)

    (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7))



                             and in Maple notation



25/144*(1891421*n^8+26671882*n^7+162583042*n^6+559578840*n^5+1189594689*n^4+
1599898238*n^3+1329740928*n^2+624648960*n+127008000)*(5*n+2)*(5*n+3)*(5*n+4)*(5
*n+1)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+
5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (4996391 n  + 69675930 n  + 418266896 n  + 1412236026 n  + 2934877253 n

                   3               2
     + 3846857436 n  + 3108402228 n  + 1417148640 n + 279417600) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(72 (2 n + 3)

    (5 + 2 n) (n + 2) (n + 3) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8)

    (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/72*(4996391*n^8+69675930*n^7+418266896*n^6+1412236026*n^5+2934877253*n^4+
3846857436*n^3+3108402228*n^2+1417148640*n+279417600)*(5*n+1)*(5*n+2)*(5*n+3)*(
5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2
)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            8             7             6              5              4
25 (702581 n  + 10164032 n  + 63550452 n  + 224388965 n  + 489551254 n

                  3              2
     + 676021913 n  + 577177593 n  + 278607210 n + 58212000) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)/(24 (2 n + 3)

    (5 + 2 n) (n + 2) (n + 3) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8)

    (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/24*(702581*n^8+10164032*n^7+63550452*n^6+224388965*n^5+489551254*n^4+
676021913*n^3+577177593*n^2+278607210*n+58212000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+
1)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/
(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4              3
25 (2172453 n  + 28825792 n  + 161639400 n  + 496345750 n  + 900940467 n

                  2
     + 966041338 n  + 566098800 n + 139708800) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)/(216 (2 n + 3) (5 + 2 n)

    (n + 2) (n + 3) (5 + 4 n) (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n)

    (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/216*(2172453*n^7+28825792*n^6+161639400*n^5+496345750*n^4+900940467*n^3+
966041338*n^2+566098800*n+139708800)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5
*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(5+4*n)/(4*n+9)/(3*n+2)/(3*n+
5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8              7               6               5                4
25 (19735209 n  + 277947716 n  + 1687881316 n  + 5775471058 n  + 12187166843 n

                    3                2
     + 16252697802 n  + 13388459976 n  + 6234284880 n + 1257379200) (5 n + 1)

                                                                          /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (
                                                                        /

                                   2
    864 (2 n + 3) (5 + 2 n) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (3 n + 2)

    (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



25/864*(19735209*n^8+277947716*n^7+1687881316*n^6+5775471058*n^5+12187166843*n^
4+16252697802*n^3+13388459976*n^2+6234284880*n+1257379200)*(5*n+1)*(5*n+2)*(5*n
+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+3)/(5+2*n)/(n+3)^2/(n+2)/(5+4*n)
/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+
11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6              5               4
125 (1801125 n  + 26336648 n  + 166648464 n  + 596224922 n  + 1319614791 n

                   3               2
     + 1850620934 n  + 1606079028 n  + 788569128 n + 167650560) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                                   2
    864 (2 n + 3) (5 + 2 n) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (3 n + 2)

    (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



125/864*(1801125*n^8+26336648*n^7+166648464*n^6+596224922*n^5+1319614791*n^4+
1850620934*n^3+1606079028*n^2+788569128*n+167650560)*(5*n+2)*(5*n+3)*(5*n+4)*(5
*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+3)^2/(n+2)/(5+4*n)/(4*
n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8              7               6               5                4
25 (17695437 n  + 273055844 n  + 1825709424 n  + 6906180410 n  + 16157984823 n

                    3                2
     + 23928555626 n  + 21887286396 n  + 11294866440 n + 2514758400) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                                   2
    864 (2 n + 3) (5 + 2 n) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (3 n + 2)

    (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



25/864*(17695437*n^8+273055844*n^7+1825709424*n^6+6906180410*n^5+16157984823*n^
4+23928555626*n^3+21887286396*n^2+11294866440*n+2514758400)*(5*n+2)*(5*n+3)*(5*
n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+3)^2/(n+2)/(5+4
*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4
*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8              7               6                5
25 (27715257 n  + 437800994 n  + 2994800844 n  + 11580164210 n

                    4                3                2
     + 27664752303 n  + 41777568476 n  + 38908413276 n  + 20409056640 n

     + 4610390400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                                 2
    (5 n + 7) (5 n + 6)  /  (864 (2 n + 3) (5 + 2 n) (n + 3)  (n + 2) (5 + 4 n)
                        /

    (4 n + 9) (3 n + 2) (3 n + 5) (3 n + 8) (1 + 3 n) (4 + 3 n) (7 + 3 n)

    (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/864*(27715257*n^8+437800994*n^7+2994800844*n^6+11580164210*n^5+27664752303*n
^4+41777568476*n^3+38908413276*n^2+20409056640*n+4610390400)*(5*n+2)*(5*n+3)*(5
*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+3)^2/(n+2)/(5+
4*n)/(4*n+9)/(3*n+2)/(3*n+5)/(3*n+8)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(
4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   7              6              5
    (5 n + 2) (5 n + 1) (12651951 n  + 160981579 n  + 863725317 n

                   4               3               2
     + 2534528293 n  + 4396513536 n  + 4512703564 n  + 2540347920 n + 605404800

        /
    )  /  (864 (4 n + 13) (4 n + 9) (5 + 4 n) (3 n + 10) (7 + 3 n) (4 + 3 n)
      /

    (1 + 3 n) (3 n + 8) (3 n + 5) (3 n + 2) (4 n + 11) (4 n + 7) (n + 2)

           2
    (n + 3)  (5 + 2 n))



                             and in Maple notation



25/864*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
12651951*n^7+160981579*n^6+863725317*n^5+2534528293*n^4+4396513536*n^3+
4512703564*n^2+2540347920*n+605404800)/(4*n+13)/(4*n+9)/(5+4*n)/(3*n+10)/(7+3*n
)/(4+3*n)/(1+3*n)/(3*n+8)/(3*n+5)/(3*n+2)/(4*n+11)/(4*n+7)/(n+2)/(n+3)^2/(5+2*n
)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8              7              6               5               4
25 (10221067 n  + 151062699 n  + 967068899 n  + 3503741620 n  + 7859810133 n

                    3               2
     + 11180203031 n  + 9847209861 n  + 4908443490 n + 1059458400) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

    144 (1 + 3 n) (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8)

    (5 + 4 n) (3 n + 2) (5 + 2 n) (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3)

           2
    (n + 3) )



                             and in Maple notation



25/144*(10221067*n^8+151062699*n^7+967068899*n^6+3503741620*n^5+7859810133*n^4+
11180203031*n^3+9847209861*n^2+4908443490*n+1059458400)*(5*n+2)*(5*n+3)*(5*n+4)
*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(1+3*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*
n)/(3*n+10)/(3*n+8)/(5+4*n)/(3*n+2)/(5+2*n)/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)/(n
+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6               5               4
125 (5284326 n  + 82426205 n  + 557371527 n  + 2133231467 n  + 5051532567 n

                   3               2
     + 7573306952 n  + 7013367276 n  + 3663846720 n + 825552000) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /
    (5 n + 6)  /  (864 (1 + 3 n) (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n)
              /

    (3 n + 10) (3 n + 8) (5 + 4 n) (3 n + 2) (5 + 2 n) (4 n + 13) (4 n + 9)

                               2
    (4 n + 7) (2 n + 3) (n + 3)  (n + 2))



                             and in Maple notation



125/864*(5284326*n^8+82426205*n^7+557371527*n^6+2133231467*n^5+5051532567*n^4+
7573306952*n^3+7013367276*n^2+3663846720*n+825552000)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(1+3*n)/(4+3*n)/(4*n+11)/(3*n+5
)/(7+3*n)/(3*n+10)/(3*n+8)/(5+4*n)/(3*n+2)/(5+2*n)/(4*n+13)/(4*n+9)/(4*n+7)/(2*
n+3)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9               8                7                6
25 (121484589 n  + 2231397686 n  + 18069172874 n  + 84619644236 n

                     5                 4                 3                 2
     + 252408594401 n  + 496959509414 n  + 645304053776 n  + 532388483664 n

     + 252954833760 n + 52670217600) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1)

                                              /
    (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (432 (1 + 3 n) (4 + 3 n)
                                            /

    (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (5 + 4 n) (3 n + 2)

                                                              2
    (5 + 2 n) (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3) (n + 3)  (n + 2))



                             and in Maple notation



25/432*(121484589*n^9+2231397686*n^8+18069172874*n^7+84619644236*n^6+
252408594401*n^5+496959509414*n^4+645304053776*n^3+532388483664*n^2+
252954833760*n+52670217600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*
n+7)*(5*n+6)/(1+3*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(5+4*n)/
(3*n+2)/(5+2*n)/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 4, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                             9               8                7
25 (5 n + 2) (5 n + 3) (5 n + 4) (117943402 n  + 2191432563 n  + 17936320682 n

                    6                 5                 4                 3
     + 84830785428 n  + 255339444248 n  + 506890357557 n  + 663116716948 n

                     2
     + 550740855492 n  + 263222988480 n + 55091836800) (5 n + 1) (5 n + 9)

                                    /
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (288 (1 + 3 n) (4 + 3 n) (4 n + 11)
                                  /

    (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (5 + 4 n) (3 n + 2) (5 + 2 n)

                                                    2
    (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3) (n + 3)  (n + 2))



                             and in Maple notation



25/288*(5*n+2)*(5*n+3)*(5*n+4)*(117943402*n^9+2191432563*n^8+17936320682*n^7+
84830785428*n^6+255339444248*n^5+506890357557*n^4+663116716948*n^3+550740855492
*n^2+263222988480*n+55091836800)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(1+3*n
)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(5+4*n)/(3*n+2)/(5+2*n)/(4*
n+13)/(4*n+9)/(4*n+7)/(2*n+3)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4         3         2
700 (384 n  + 2511 n  + 5839 n  + 5666 n + 1980) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6)/(27 (n + 2) (n + 3) (3 n + 2) (3 n + 5)

    (3 n + 8) (3 n + 11) (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



700/27*(384*n^4+2511*n^3+5839*n^2+5666*n+1980)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*
(1+n)*(5*n+6)/(n+2)/(n+3)/(3*n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(1+3*n)/(4+3*n)/(7+3
*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7                6
25 (42093741 n  + 698837985 n  + 5082271406 n  + 21242844608 n

                    5                4                 3                2
     + 56228540979 n  + 97743020455 n  + 111608904754 n  + 80762874072 n

     + 33632200800 n + 6147187200) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (1 + n) (5 n + 6) (5 n + 7)/(144 (2 n + 3) (5 + 2 n) (n + 2) (n + 3)

    (4 n + 7) (4 n + 11) (3 n + 2) (3 n + 5) (3 n + 8) (3 n + 11) (1 + 3 n)

    (4 + 3 n) (7 + 3 n) (3 n + 10) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/144*(42093741*n^9+698837985*n^8+5082271406*n^7+21242844608*n^6+56228540979*n
^5+97743020455*n^4+111608904754*n^3+80762874072*n^2+33632200800*n+6147187200)*(
5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+3)/(5+2*n)/(n+2)/(n+3
)/(4*n+7)/(4*n+11)/(3*n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(1+3*n)/(4+3*n)/(7+3*n)/(3*
n+10)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                      9              8
25 (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (16343093 n  + 279527595 n

                   7               6                5                4
     + 2100900622 n  + 9105953610 n  + 25082405317 n  + 45536028915 n

                    3                2
     + 54492861008 n  + 41457885720 n  + 18198999720 n + 3512678400) (5 n + 8)

    (5 n + 7) (5 n + 6)/(144 (2 n + 3) (5 + 2 n) (n + 2) (n + 3) (4 n + 7)

    (4 n + 11) (3 n + 2) (3 n + 5) (3 n + 8) (3 n + 11) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (3 n + 10) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/144*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(16343093*n^9+279527595*n^8+2100900622*n
^7+9105953610*n^6+25082405317*n^5+45536028915*n^4+54492861008*n^3+41457885720*n
^2+18198999720*n+3512678400)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)/(n+3
)/(4*n+7)/(4*n+11)/(3*n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(1+3*n)/(4+3*n)/(7+3*n)/(3*
n+10)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               8              7               6               5               4
175 (13707253 n  + 193685948 n  + 1179097370 n  + 4041228568 n  + 8534976789 n

                    3               2
     + 11383545660 n  + 9372496092 n  + 4359809520 n + 878169600) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)/(576

    (2 n + 3) (5 + 2 n) (n + 2) (n + 3) (4 n + 7) (4 n + 11) (3 n + 2)

    (3 n + 5) (3 n + 8) (3 n + 11) (1 + 3 n) (4 + 3 n) (7 + 3 n) (3 n + 10)

    (5 + 4 n) (4 n + 9))



                             and in Maple notation



175/576*(13707253*n^8+193685948*n^7+1179097370*n^6+4041228568*n^5+8534976789*n^
4+11383545660*n^3+9372496092*n^2+4359809520*n+878169600)*(5*n+1)*(5*n+2)*(5*n+3
)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(4*n+7)/(4*
n+11)/(3*n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(5+4*n)
/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8              7               6                5
25 (42381795 n  + 621143664 n  + 3936345946 n  + 14094461460 n

                    4                3                2
     + 31199856535 n  + 43738034556 n  + 37929599404 n  + 18605178240 n

     + 3951763200) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

    (5 n + 7) (5 n + 6)/(576 (2 n + 3) (5 + 2 n) (n + 2) (n + 3) (4 n + 7)

    (4 n + 11) (3 n + 2) (3 n + 5) (3 n + 8) (3 n + 11) (1 + 3 n) (4 + 3 n)

    (7 + 3 n) (3 n + 10) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/576*(42381795*n^8+621143664*n^7+3936345946*n^6+14094461460*n^5+31199856535*n
^4+43738034556*n^3+37929599404*n^2+18605178240*n+3951763200)*(5*n+2)*(5*n+3)*(5
*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)/(5+2*n)/(n+2)/(n+3)/(4*n+
7)/(4*n+11)/(3*n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(1+3*n)/(4+3*n)/(7+3*n)/(3*n+10)/(
5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               7              6               5               4               3
125 (15487667 n  + 208250478 n  + 1183662550 n  + 3685057920 n  + 6783104823 n

                   2
     + 7376911122 n  + 4384923840 n + 1097712000) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)/(576 (3 n + 11) (1 + 3 n)

    (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (5 + 4 n)

    (3 n + 2) (5 + 2 n) (4 n + 9) (4 n + 7) (2 n + 3) (n + 3))



                             and in Maple notation



125/576*(15487667*n^7+208250478*n^6+1183662550*n^5+3685057920*n^4+6783104823*n^
3+7376911122*n^2+4384923840*n+1097712000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+
9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+11)/(1+3*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3
*n+10)/(3*n+8)/(5+4*n)/(3*n+2)/(5+2*n)/(4*n+9)/(4*n+7)/(2*n+3)/(n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    9               8                7
    (5 n + 2) (5 n + 1) (523874202 n  + 9134257677 n  + 69686705894 n

                     6                 5                  4                  3
     + 305427936720 n  + 847925477912 n  + 1547230262919 n  + 1856982650360 n

                      2                                    /
     + 1414667718876 n  + 621192821040 n + 119870150400)  /  (1728 (4 n + 11)
                                                         /

    (4 n + 7) (3 n + 10) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 11) (3 n + 8)

                                                                      2
    (3 n + 5) (3 n + 2) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 2) (n + 3)

    (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/1728*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
523874202*n^9+9134257677*n^8+69686705894*n^7+305427936720*n^6+847925477912*n^5+
1547230262919*n^4+1856982650360*n^3+1414667718876*n^2+621192821040*n+
119870150400)/(4*n+11)/(4*n+7)/(3*n+10)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+11)/(3*n+8
)/(3*n+5)/(3*n+2)/(4*n+13)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9               8                7                 6
25 (208098457 n  + 3741440083 n  + 29522853502 n  + 134233385858 n

                     5                 4                 3                 2
     + 387685985273 n  + 737796331707 n  + 925390323608 n  + 737726741232 n

     + 339172625880 n + 68497228800) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1)

                                              /
    (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (288 (3 n + 11) (1 + 3 n)
                                            /

    (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (5 + 4 n)

                                                                        2
    (3 n + 2) (5 + 2 n) (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3) (n + 3) )



                             and in Maple notation



25/288*(208098457*n^9+3741440083*n^8+29522853502*n^7+134233385858*n^6+
387685985273*n^5+737796331707*n^4+925390323608*n^3+737726741232*n^2+
339172625880*n+68497228800)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*
n+7)*(5*n+6)/(3*n+11)/(1+3*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)
/(5+4*n)/(3*n+2)/(5+2*n)/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
625 (20820330 n  + 390979503 n  + 3228000040 n  + 15376288062 n

                    5                4                 3                 2
     + 46557922618 n  + 92895588543 n  + 122078871844 n  + 101824774644 n

     + 48873890736 n + 10274584320) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11)

                                                        /
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1728 (4 n + 11)
                                                      /

    (4 n + 7) (3 n + 10) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 11) (3 n + 8)

                                                                      2
    (3 n + 5) (3 n + 2) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 2) (n + 3)

    (5 + 2 n) (2 n + 3))



                             and in Maple notation



625/1728*(20820330*n^9+390979503*n^8+3228000040*n^7+15376288062*n^6+46557922618
*n^5+92895588543*n^4+122078871844*n^3+101824774644*n^2+48873890736*n+
10274584320)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(
5*n+6)/(4*n+11)/(4*n+7)/(3*n+10)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+11)/(3*n+8)/(3*n+
5)/(3*n+2)/(4*n+13)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        10                9
25 (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (2218721559 n   + 47742291354 n

                     8                  7                  6
     + 458273724821 n  + 2583260639984 n  + 9466165149773 n

                       5                   4                   3
     + 23551131457946 n  + 40265575546259 n  + 46683329724236 n

                       2
     + 35098464072948 n  + 15438648859920 n + 3013878067200) (5 n + 9)

                                    /
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (864 (4 n + 11) (4 n + 7) (3 n + 10)
                                  /

    (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (3 n + 2)

                                                  2
    (4 n + 13) (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/864*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(2218721559*n^10+47742291354*n^9+
458273724821*n^8+2583260639984*n^7+9466165149773*n^6+23551131457946*n^5+
40265575546259*n^4+46683329724236*n^3+35098464072948*n^2+15438648859920*n+
3013878067200)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(4*n+11)/(4*n+7)/(3*n+10)/(7+3*n
)/(4+3*n)/(1+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+2)/(4*n+13)/(4*n+9)/(5+4*n)/(n+
2)/(n+3)^2/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    9               8                7
    (5 n + 2) (5 n + 1) (550501797 n  + 9668864473 n  + 74381260407 n

                     6                 5                  4                  3
     + 329087209158 n  + 923319781813 n  + 1704730152287 n  + 2072606028983 n

                      2                                    /
     + 1601149998402 n  + 713601476280 n + 139848508800)  /  (576 (4 n + 11)
                                                         /

    (4 n + 7) (3 n + 10) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 11) (3 n + 8)

                                                              2
    (3 n + 5) (3 n + 2) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 3)  (7 + 2 n)

    (5 + 2 n) (2 n + 3))



                             and in Maple notation



25/576*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
550501797*n^9+9668864473*n^8+74381260407*n^7+329087209158*n^6+923319781813*n^5+
1704730152287*n^4+2072606028983*n^3+1601149998402*n^2+713601476280*n+
139848508800)/(4*n+11)/(4*n+7)/(3*n+10)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+11)/(3*n+8
)/(3*n+5)/(3*n+2)/(4*n+13)/(4*n+9)/(5+4*n)/(n+3)^2/(7+2*n)/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7               6
625 (11063165 n  + 200449077 n  + 1595185631 n  + 7320367308 n

                    5                4                3                2
     + 21354538595 n  + 41074727529 n  + 52099968393 n  + 42021089718 n

     + 19550533464 n + 3995671680) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (576 (4 n + 11)
                                                      /

    (4 n + 7) (3 n + 10) (7 + 3 n) (4 + 3 n) (1 + 3 n) (3 n + 11) (3 n + 8)

                                                              2
    (3 n + 5) (3 n + 2) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 3)  (7 + 2 n)

    (5 + 2 n) (2 n + 3))



                             and in Maple notation



625/576*(11063165*n^9+200449077*n^8+1595185631*n^7+7320367308*n^6+21354538595*n
^5+41074727529*n^4+52099968393*n^3+42021089718*n^2+19550533464*n+3995671680)*(5
*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(4*n+11)
/(4*n+7)/(3*n+10)/(7+3*n)/(4+3*n)/(1+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(3*n+2)/(4*n
+13)/(4*n+9)/(5+4*n)/(n+3)^2/(7+2*n)/(5+2*n)/(2*n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
625 (14966031 n  + 283167033 n  + 2356620712 n  + 11320268714 n

                    5                4                3                2
     + 34578491347 n  + 69621230681 n  + 92344073134 n  + 77747170820 n

     + 37666658568 n + 7991343360) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (1728
                                                                 /

    (3 n + 11) (1 + 3 n) (7 + 2 n) (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n)

    (3 n + 10) (3 n + 8) (5 + 4 n) (3 n + 2) (5 + 2 n) (4 n + 13) (4 n + 9)

                               2
    (4 n + 7) (2 n + 3) (n + 3)  (n + 2))



                             and in Maple notation



625/1728*(14966031*n^9+283167033*n^8+2356620712*n^7+11320268714*n^6+34578491347
*n^5+69621230681*n^4+92344073134*n^3+77747170820*n^2+37666658568*n+7991343360)*
(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+
9)/(3*n+11)/(1+3*n)/(7+2*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(
5+4*n)/(3*n+2)/(5+2*n)/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                11                 10                  9                   8
25 (6076218537 n   + 147346000664 n   + 1612622681805 n  + 10511259469860 n

                       7                    6                    5
     + 45321495851031 n  + 135671385494352 n  + 287587749319335 n

                        4                    3                    2
     + 431427543271540 n  + 448586897161932 n  + 307661663049984 n

     + 125157227247360 n + 22855242009600) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (576 (3 n + 11)
                                                      /

    (1 + 3 n) (7 + 2 n) (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10)

    (3 n + 8) (5 + 4 n) (3 n + 2) (5 + 2 n) (4 n + 13) (4 n + 9) (4 n + 7)

                     2
    (2 n + 3) (n + 3)  (n + 2))



                             and in Maple notation



25/576*(6076218537*n^11+147346000664*n^10+1612622681805*n^9+10511259469860*n^8+
45321495851031*n^7+135671385494352*n^6+287587749319335*n^5+431427543271540*n^4+
448586897161932*n^3+307661663049984*n^2+125157227247360*n+22855242009600)*(5*n+
2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(3*n+11)/(1+3*n)/(7+
2*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(5+4*n)/(3*n+2)/(5+2*n)/
(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n, n + 5, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



                11                 10                  9                   8
25 (9477958061 n   + 232980448482 n   + 2583354738935 n  + 17050178096400 n

                       7                    6                    5
     + 74394083193663 n  + 225219429959946 n  + 482483982623785 n

                        4                    3                    2
     + 730998328521300 n  + 767084290561876 n  + 530569074801072 n

     + 217507208798880 n + 39996673516800) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (576 (3 n + 11)
                                                      /

    (1 + 3 n) (7 + 2 n) (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10)

    (3 n + 8) (5 + 4 n) (3 n + 2) (5 + 2 n) (4 n + 13) (4 n + 9) (4 n + 7)

                     2
    (2 n + 3) (n + 3)  (n + 2))



                             and in Maple notation



25/576*(9477958061*n^11+232980448482*n^10+2583354738935*n^9+17050178096400*n^8+
74394083193663*n^7+225219429959946*n^6+482483982623785*n^5+730998328521300*n^4+
767084290561876*n^3+530569074801072*n^2+217507208798880*n+39996673516800)*(5*n+
2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(3*n+11)/(1+3*n)/(7+
2*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(5+4*n)/(3*n+2)/(5+2*n)/
(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



        5         4          3          2
5 (896 n  + 6088 n  + 14947 n  + 16328 n  + 7731 n + 1260) (5 n + 1) (5 n + 2)

                          /
    (5 n + 3) (5 n + 4)  /  (16 (4 n + 1) (5 + 4 n) (4 n + 3) (5 + 2 n)
                        /

                       2          2
    (7 + 2 n) (2 n + 1)  (2 n + 3) )



                             and in Maple notation



5/16*(896*n^5+6088*n^4+14947*n^3+16328*n^2+7731*n+1260)*(5*n+1)*(5*n+2)*(5*n+3)
*(5*n+4)/(4*n+1)/(5+4*n)/(4*n+3)/(5+2*n)/(7+2*n)/(2*n+1)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          6           5           4            3           2
5 (35248 n  + 275868 n  + 831808 n  + 1234707 n  + 950575 n  + 361074 n + 52920

                                                /
    ) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)  /  (64 (4 n + 1) (5 + 4 n)
                                              /

                                 2          2
    (5 + 2 n) (7 + 2 n) (2 n + 1)  (2 n + 3)  (4 n + 3) (4 n + 7))



                             and in Maple notation



5/64*(35248*n^6+275868*n^5+831808*n^4+1234707*n^3+950575*n^2+361074*n+52920)*(5
*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(4*n+1)/(5+4*n)/(5+2*n)/(7+2*n)/(2*n+1)^2/(2*n+3)
^2/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          6           5            4            3            2
5 (53936 n  + 445404 n  + 1440848 n  + 2337543 n  + 2002319 n  + 855330 n

                                                         /
     + 141120) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)  /  (64 (4 n + 1)
                                                       /

                                           2          2
    (5 + 4 n) (5 + 2 n) (7 + 2 n) (2 n + 1)  (2 n + 3)  (4 n + 3) (4 n + 7))



                             and in Maple notation



5/64*(53936*n^6+445404*n^5+1440848*n^4+2337543*n^3+2002319*n^2+855330*n+141120)
*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(4*n+1)/(5+4*n)/(5+2*n)/(7+2*n)/(2*n+1)^2/(2*n
+3)^2/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                   7           6            5
15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (50264 n  + 452482 n  + 1646337 n

                4            3            2                       /
     + 3146884 n  + 3423987 n  + 2127862 n  + 700504 n + 94080)  /  (128
                                                                /

                               2          2
    (n + 2) (4 n + 7) (2 n + 3)  (2 n + 1)  (7 + 2 n) (5 + 2 n) (4 n + 3)

    (5 + 4 n) (4 n + 1))



                             and in Maple notation



15/128*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(50264*n^7+452482*n^6+1646337*n^5+
3146884*n^4+3423987*n^3+2127862*n^2+700504*n+94080)/(n+2)/(4*n+7)/(2*n+3)^2/(2*
n+1)^2/(7+2*n)/(5+2*n)/(4*n+3)/(5+4*n)/(4*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                   7            6             5
5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (274408 n  + 2652926 n  + 10523731 n

                 4             3             2                          /
     + 22244570 n  + 27074347 n  + 18950594 n  + 7033824 n + 1058400)  /  (128
                                                                      /

                               2          2
    (n + 2) (4 n + 7) (2 n + 3)  (2 n + 1)  (7 + 2 n) (5 + 2 n) (4 n + 3)

    (5 + 4 n) (4 n + 1))



                             and in Maple notation



5/128*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(274408*n^7+2652926*n^6+10523731*n^5+
22244570*n^4+27074347*n^3+18950594*n^2+7033824*n+1058400)/(n+2)/(4*n+7)/(2*n+3)
^2/(2*n+1)^2/(7+2*n)/(5+2*n)/(4*n+3)/(5+4*n)/(4*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                   6           5            4
15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (45704 n  + 396518 n  + 1358531 n

                3            2                                  /
     + 2351204 n  + 2161637 n  + 994406 n + 176400) (4 + 3 n)  /  (128 (n + 2)
                                                              /

                       2          2
    (4 n + 7) (2 n + 3)  (2 n + 1)  (7 + 2 n) (5 + 2 n) (4 n + 3) (5 + 4 n)

    (4 n + 1))



                             and in Maple notation



15/128*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(45704*n^6+396518*n^5+1358531*n^4+
2351204*n^3+2161637*n^2+994406*n+176400)*(4+3*n)/(n+2)/(4*n+7)/(2*n+3)^2/(2*n+1
)^2/(7+2*n)/(5+2*n)/(4*n+3)/(5+4*n)/(4*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5            4            3            2
25 (237592 n  + 1742450 n  + 4927319 n  + 7001827 n  + 5328744 n  + 2066148 n

                                                                 /
     + 317520) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)  /  (128
                                                               /

                                 2          2
    (5 + 2 n) (7 + 2 n) (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9)

    (4 n + 3) (4 n + 7))



                             and in Maple notation



25/128*(237592*n^6+1742450*n^5+4927319*n^4+7001827*n^3+5328744*n^2+2066148*n+
317520)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(5+2*n)/(7+2*n)/(2*n+1)^2/(2*n+3)
^2/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5          4          3           2
525 (4600 n  + 32306 n  + 84261 n  + 102958 n  + 58619 n + 12096) (5 n + 1)

                                                      /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)  /  (128 (5 + 2 n)
                                                    /

                       2          2
    (7 + 2 n) (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3)

    (4 n + 7))



                             and in Maple notation



525/128*(4600*n^5+32306*n^4+84261*n^3+102958*n^2+58619*n+12096)*(5*n+1)*(5*n+2)
*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(5+2*n)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4
*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5             4             3
35 (611336 n  + 5965182 n  + 24035807 n  + 51993960 n  + 65211179 n

                 2
     + 47291268 n  + 18249588 n + 2857680) (5 n + 1) (5 n + 2) (5 n + 3)

                /                                   2          2
    (5 n + 4)  /  (128 (5 + 2 n) (7 + 2 n) (2 n + 1)  (2 n + 3)  (4 n + 1)
              /

    (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



35/128*(611336*n^7+5965182*n^6+24035807*n^5+51993960*n^4+65211179*n^3+47291268*
n^2+18249588*n+2857680)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(5+2*n)/(7+2*n)/(2*n+1)
^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5             4             3             2
35 (451636 n  + 3856113 n  + 13029205 n  + 22354485 n  + 20501539 n

                                                                      /
     + 9464862 n + 1693440) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)  /  (128
                                                                    /

                                           2
    (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 1)  (4 n + 1) (5 + 4 n) (4 n + 9)

    (4 n + 3) (4 n + 7))



                             and in Maple notation



35/128*(451636*n^6+3856113*n^5+13029205*n^4+22354485*n^3+20501539*n^2+9464862*n
+1693440)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+1)^2/(4*
n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5             4             3
25 (353912 n  + 3408250 n  + 13397147 n  + 27939010 n  + 33486653 n

                 2
     + 23107780 n  + 8486688 n + 1270080) (5 n + 1) (5 n + 2) (5 n + 3)

                                  /                                 2
    (5 n + 4) (1 + n) (5 n + 6)  /  (256 (n + 2) (7 + 2 n) (2 n + 1)
                                /

             2          2
    (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



25/256*(353912*n^7+3408250*n^6+13397147*n^5+27939010*n^4+33486653*n^3+23107780*
n^2+8486688*n+1270080)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(n+2)/(7+2
*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6            5             4             3
25 (155896 n  + 1611322 n  + 6881827 n  + 15758170 n  + 20883979 n

                 2
     + 15969838 n  + 6476568 n + 1058400) (5 n + 1) (5 n + 2) (5 n + 3)

                                    /                                 2
    (5 n + 4) (5 n + 6) (5 n + 7)  /  (256 (n + 2) (7 + 2 n) (2 n + 1)
                                  /

             2          2
    (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



25/256*(155896*n^7+1611322*n^6+6881827*n^5+15758170*n^4+20883979*n^3+15969838*n
^2+6476568*n+1058400)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(n+2)/(7+
2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                      9              8
15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (12819880 n  + 175111286 n

                   7               6               5                4
     + 1041659387 n  + 3541712277 n  + 7580630645 n  + 10578179169 n

                   3               2                               /
     + 9601385608 n  + 5446272628 n  + 1741864320 n + 237081600)  /  (256
                                                                 /

                               2          2          2
    (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



15/256*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(12819880*n^9+175111286*n^8+1041659387*n
^7+3541712277*n^6+7580630645*n^5+10578179169*n^4+9601385608*n^3+5446272628*n^2+
1741864320*n+237081600)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+
4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                     9              8
5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (66840920 n  + 942653274 n

                   7                6                5                4
     + 5791616193 n  + 20334035433 n  + 44908357605 n  + 64583649921 n

                    3                2                                 /
     + 60316569262 n  + 35133827712 n  + 11511092880 n + 1600300800)  /  (256
                                                                     /

                               2          2          2
    (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



5/256*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(66840920*n^9+942653274*n^8+5791616193*n^
7+20334035433*n^6+44908357605*n^5+64583649921*n^4+60316569262*n^3+35133827712*n
^2+11511092880*n+1600300800)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1
)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                      9              8
15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (32563688 n  + 466943174 n

                   7                6                5                4
     + 2916779867 n  + 10408510641 n  + 23352839417 n  + 34095697701 n

                    3                2                               /
     + 32302320708 n  + 19069453684 n  + 6325164720 n + 889056000)  /  (256
                                                                   /

                               2          2          2
    (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



15/256*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(32563688*n^9+466943174*n^8+2916779867*n
^7+10408510641*n^6+23352839417*n^5+34095697701*n^4+32302320708*n^3+19069453684*
n^2+6325164720*n+889056000)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)
/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                   35 (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)
                   ------------------------------------------
                   8 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n)



                             and in Maple notation



35/8*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                 3         2
15 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (309 n  + 1149 n  + 1402 n + 560)

       /                                2
      /  (64 (n + 2) (2 n + 1) (2 n + 3)  (7 + 2 n) (5 + 2 n) (5 + 4 n))
     /



                             and in Maple notation



15/64*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(309*n^3+1149*n^2+1402*n+560)/(n+2)/(2*n+
1)/(2*n+3)^2/(7+2*n)/(5+2*n)/(5+4*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (11 n + 16) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)/(64

                               2
    (n + 2) (2 n + 1) (2 n + 3)  (7 + 2 n) (5 + 2 n) (5 + 4 n))



                             and in Maple notation



25/64*(11*n+16)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(n+2)/(2*n+1)/(
2*n+3)^2/(7+2*n)/(5+2*n)/(5+4*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



        2
75 (37 n  + 113 n + 84) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)

                          /                                  2
    (5 n + 6) (5 n + 7)  /  (32 (5 + 2 n) (7 + 2 n) (2 n + 3)  (2 n + 1)
                        /

    (n + 2) (4 n + 7) (5 + 4 n) (4 n + 9))



                             and in Maple notation



75/32*(37*n^2+113*n+84)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(
5+2*n)/(7+2*n)/(2*n+3)^2/(2*n+1)/(n+2)/(4*n+7)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
25 (398 n  + 1860 n  + 2899 n + 1512) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                          /                                  2
    (5 n + 6) (5 n + 7)  /  (64 (5 + 2 n) (7 + 2 n) (2 n + 3)  (2 n + 1)
                        /

    (n + 2) (4 n + 7) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/64*(398*n^3+1860*n^2+2899*n+1512)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5
*n+7)/(5+2*n)/(7+2*n)/(2*n+3)^2/(2*n+1)/(n+2)/(4*n+7)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
25 (287 n  + 1410 n  + 2308 n + 1260) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                          /                                  2
    (5 n + 6) (5 n + 7)  /  (32 (5 + 2 n) (7 + 2 n) (2 n + 3)  (2 n + 1)
                        /

    (n + 2) (4 n + 7) (5 + 4 n) (4 n + 9))



                             and in Maple notation



25/32*(287*n^3+1410*n^2+2308*n+1260)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5
*n+7)/(5+2*n)/(7+2*n)/(2*n+3)^2/(2*n+1)/(n+2)/(4*n+7)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          3         2
25 (1562 n  + 6933 n  + 10216 n + 5040) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                                  /                         2          2
    (1 + n) (5 n + 6) (5 n + 7)  /  (128 (7 + 2 n) (2 n + 3)  (5 + 2 n)
                                /

    (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7))



                             and in Maple notation



25/128*(1562*n^3+6933*n^2+10216*n+5040)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(
5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(5+4*n)/(4*n+9)/(4*n+7
)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4          3          2
25 (3154 n  + 19915 n  + 47345 n  + 50294 n + 20160) (5 n + 1) (5 n + 2)

                                              /                         2
    (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)  /  (128 (7 + 2 n) (2 n + 3)
                                            /

             2
    (5 + 2 n)  (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7))



                             and in Maple notation



25/128*(3154*n^4+19915*n^3+47345*n^2+50294*n+20160)*(5*n+1)*(5*n+2)*(5*n+3)*(5*
n+4)*(5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(5+4*n)/(4*n+9)/
(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          3         2
25 (1090 n  + 5643 n  + 9773 n + 5670) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                                    /                         2          2
    (5 n + 6) (5 n + 7) (5 n + 8)  /  (128 (7 + 2 n) (2 n + 3)  (5 + 2 n)
                                  /

    (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7))



                             and in Maple notation



25/128*(1090*n^3+5643*n^2+9773*n+5670)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*
(5*n+7)*(5*n+8)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(5+4*n)/(4*n+9)/(4*n+
7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                              3          2
75 (5 n + 6) (5 n + 7) (2582 n  + 12821 n  + 21249 n + 11760) (5 n + 1)

                                    /                         2          2
    (5 n + 2) (5 n + 3) (5 n + 4)  /  (128 (7 + 2 n) (2 n + 3)  (5 + 2 n)
                                  /

    (2 n + 1) (5 + 4 n) (4 n + 9) (4 n + 7))



                             and in Maple notation



75/128*(5*n+6)*(5*n+7)*(2582*n^3+12821*n^2+21249*n+11760)*(5*n+1)*(5*n+2)*(5*n+
3)*(5*n+4)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(5+4*n)/(4*n+9)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           4          3           2
25 (10534 n  + 63377 n  + 143363 n  + 144976 n + 55440) (5 n + 1) (5 n + 2)

                                                      /
    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (128 (7 + 2 n)
                                                    /

             2          2
    (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



25/128*(10534*n^4+63377*n^3+143363*n^2+144976*n+55440)*(5*n+1)*(5*n+2)*(5*n+3)*
(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(5+4*n)
/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4         3          2
125 (926 n  + 6049 n  + 14929 n  + 16526 n + 6930) (5 n + 1) (5 n + 2)

                                                        /
    (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8)  /  (128 (7 + 2 n)
                                                      /

             2          2
    (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



125/128*(926*n^4+6049*n^3+14929*n^2+16526*n+6930)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+
4)*(5*n+6)*(5*n+7)*(5*n+8)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(5+4*n)/(4
*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          3         2
25 (1798 n  + 9725 n  + 17657 n + 10780) (5 n + 1) (5 n + 2) (5 n + 3)

                                                        /
    (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (128 (7 + 2 n)
                                                      /

             2          2
    (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



25/128*(1798*n^3+9725*n^2+17657*n+10780)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6
)*(5*n+7)*(5*n+8)*(5*n+9)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(5+4*n)/(4*
n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5           4            3            2
25 (75934 n  + 696981 n  + 2553703 n  + 4669044 n  + 4260358 n + 1552320)

                                                                  /
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)  /  (128
                                                                /

                       2          2
    (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9)

    (4 n + 7) (4 n + 11))



                             and in Maple notation



25/128*(75934*n^5+696981*n^4+2553703*n^3+4669044*n^2+4260358*n+1552320)*(5*n+1)
*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n
+2)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, 1 + n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



            5           4            3            2
25 (106918 n  + 998007 n  + 3709900 n  + 6867543 n  + 6333112 n + 2328480)

                                                                  /
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)  /  (128
                                                                /

                       2          2
    (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (5 + 4 n) (4 n + 9)

    (4 n + 7) (4 n + 11))



                             and in Maple notation



25/128*(106918*n^5+998007*n^4+3709900*n^3+6867543*n^2+6333112*n+2328480)*(5*n+1
)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(
n+2)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        2
         5 (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (10 n  + 32 n + 21)
         -------------------------------------------------------------
           2 (n + 3) (n + 2) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1)



                             and in Maple notation



5/2*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(10*n^2+32*n+21)/(n+3)/(n+2)/(7+2*n)/(5+2*n
)/(2*n+3)/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         15 (13 n + 14) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)
         --------------------------------------------------------------
           4 (n + 3) (n + 2) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1)



                             and in Maple notation



15/4*(13*n+14)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(n+3)/(n+2)/(7+2*n)/(5+2*n
)/(2*n+3)/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         25 (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)
         --------------------------------------------------------------
           8 (n + 3) (n + 2) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1)



                             and in Maple notation



25/8*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(n+3)/(n+2)/(7+2*n)/(5+2*n
)/(2*n+3)/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



75 (4 + 3 n) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)

                                     2          2
    (5 n + 7)/(16 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3))



                             and in Maple notation



75/16*(4+3*n)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(7+2*n)/(2*
n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



        2
25 (17 n  + 53 n + 42) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6)

                /                        2          2
    (5 n + 7)  /  (16 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3)
              /

    )



                             and in Maple notation



25/16*(17*n^2+53*n+42)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(7+2*n)/
(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8)

                                     2          2
    (5 n + 9)/(16 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3))



                             and in Maple notation



25/16*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(7+2*n)/(
2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4         3          2
75 (1316 n  + 9076 n  + 23239 n  + 26261 n + 11088) (5 n + 1) (5 n + 2)

                                                      /
    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (64 (7 + 2 n)
                                                    /

             2          2
    (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (4 n + 9) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



75/64*(1316*n^4+9076*n^3+23239*n^2+26261*n+11088)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+
4)*(1+n)*(5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(4*n+9
)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
25 (332 n  + 1836 n  + 3373 n + 2079) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                                              /                        2
    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (64 (7 + 2 n) (2 n + 3)
                                            /

             2
    (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/64*(332*n^3+1836*n^2+3373*n+2079)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5
*n+7)*(5*n+8)*(5*n+9)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(4*n+9)/(
4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
75 (196 n  + 1188 n  + 2399 n + 1617) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                                              /                        2
    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (64 (7 + 2 n) (2 n + 3)
                                            /

             2
    (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/64*(196*n^3+1188*n^2+2399*n+1617)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(5
*n+7)*(5*n+8)*(5*n+9)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(4*n+9)/(
4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
25 (844 n  + 5292 n  + 11021 n + 7623) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                                              /                        2
    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (64 (7 + 2 n) (2 n + 3)
                                            /

             2
    (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/64*(844*n^3+5292*n^2+11021*n+7623)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)*(
5*n+7)*(5*n+8)*(5*n+9)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(4*n+9)/
(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



225 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                             3        2                  /
    (5 n + 2) (5 n + 1) (58 n  + 305 n  + 529 n + 308)  /  (128 (4 n + 9)
                                                       /

                                        2                    2          2
    (4 n + 11) (4 n + 7) (n + 2) (n + 3)  (2 n + 1) (5 + 2 n)  (2 n + 3)

    (7 + 2 n))



                             and in Maple notation



225/128*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
58*n^3+305*n^2+529*n+308)/(4*n+9)/(4*n+11)/(4*n+7)/(n+2)/(n+3)^2/(2*n+1)/(5+2*n
)^2/(2*n+3)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4         3          2
25 (1186 n  + 8931 n  + 25268 n  + 31929 n + 15246) (5 n + 1) (5 n + 2)

                                                                  /
    (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (128
                                                                /

                                                  2                    2
    (4 n + 9) (4 n + 11) (4 n + 7) (n + 2) (n + 3)  (2 n + 1) (5 + 2 n)

             2
    (2 n + 3)  (7 + 2 n))



                             and in Maple notation



25/128*(1186*n^4+8931*n^3+25268*n^2+31929*n+15246)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n
+4)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(4*n+9)/(4*n+11)/(4*n+7)/(n+2)/(n+3)^2/(2*n
+1)/(5+2*n)^2/(2*n+3)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4          3          2
25 (2362 n  + 19587 n  + 61046 n  + 84813 n + 44352) (5 n + 1) (5 n + 2)

                                                                  /
    (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (128
                                                                /

                                                  2                    2
    (4 n + 9) (4 n + 11) (4 n + 7) (n + 2) (n + 3)  (2 n + 1) (5 + 2 n)

             2
    (2 n + 3)  (7 + 2 n))



                             and in Maple notation



25/128*(2362*n^4+19587*n^3+61046*n^2+84813*n+44352)*(5*n+1)*(5*n+2)*(5*n+3)*(5*
n+4)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(4*n+9)/(4*n+11)/(4*n+7)/(n+2)/(n+3)^2/(2*
n+1)/(5+2*n)^2/(2*n+3)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



1125 (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (5 n + 4) (5 n + 3)

    (5 n + 2) (5 n + 1) (6 n + 13) (7 + 3 n)/(128 (4 n + 11) (4 n + 7)

                     2                    2          2
    (4 n + 9) (n + 3)  (2 n + 1) (5 + 2 n)  (2 n + 3)  (7 + 2 n))



                             and in Maple notation



1125/128*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+
1)*(6*n+13)*(7+3*n)/(4*n+11)/(4*n+7)/(4*n+9)/(n+3)^2/(2*n+1)/(5+2*n)^2/(2*n+3)^
2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 2, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4          3           2
25 (5738 n  + 50883 n  + 168634 n  + 247557 n + 135828) (5 n + 1) (5 n + 2)

                                                                  /
    (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (128
                                                                /

                                                  2                    2
    (4 n + 9) (4 n + 11) (4 n + 7) (n + 2) (n + 3)  (2 n + 1) (5 + 2 n)

             2
    (2 n + 3)  (7 + 2 n))



                             and in Maple notation



25/128*(5738*n^4+50883*n^3+168634*n^2+247557*n+135828)*(5*n+1)*(5*n+2)*(5*n+3)*
(5*n+4)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(4*n+9)/(4*n+11)/(4*n+7)/(n+2)/(n+3)^2/
(2*n+1)/(5+2*n)^2/(2*n+3)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6            5             4             3             2
5 (347161 n  + 4012406 n  + 18942953 n  + 46666684 n  + 63152316 n

     + 44447760 n + 12700800) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)/(

    72 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5)

    (3 n + 8) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



5/72*(347161*n^6+4012406*n^5+18942953*n^4+46666684*n^3+63152316*n^2+44447760*n+
12700800)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)
/(n+2)/(n+3)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5           4           3           2
25 (12769 n  + 120516 n  + 442833 n  + 791794 n  + 690088 n + 235200) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)/(24 (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/24*(12769*n^5+120516*n^4+442833*n^3+791794*n^2+690088*n+235200)*(5*n+1)*(5*n
+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(
3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          5          4           3           2
25 (3362 n  + 32930 n  + 126499 n  + 238411 n  + 220860 n + 80640) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7)/(18 (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/18*(3362*n^5+32930*n^4+126499*n^3+238411*n^2+220860*n+80640)*(5*n+1)*(5*n+2)
*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(3
*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4          3           2
25 (6671 n  + 45501 n  + 113776 n  + 124356 n + 50400) (5 n + 1) (5 n + 2)

    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(36 (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/36*(6671*n^4+45501*n^3+113776*n^2+124356*n+50400)*(5*n+1)*(5*n+2)*(5*n+3)*(5
*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(3*n+5)
/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         4         3          2
25 (893 n  + 6440 n  + 17181 n  + 20174 n + 8820) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8)/(12 (2 n + 1) (2 n + 3) (5 + 2 n)

    (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/12*(893*n^4+6440*n^3+17181*n^2+20174*n+8820)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)
*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(3*n+5)/(3
*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          3         2
25 (1697 n  + 9933 n  + 18946 n + 11760) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)/(72 (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/72*(1697*n^3+9933*n^2+18946*n+11760)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+6)
*(5*n+7)*(5*n+8)*(5*n+9)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(3*n+5)/(3
*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5             4             3             2
25 (285383 n  + 3343155 n  + 16147483 n  + 41164797 n  + 58445046 n

     + 43857720 n + 13608000) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)

                                    /                         2
    (5 n + 6) (5 n + 7) (5 n + 8)  /  (576 (7 + 2 n) (5 + 2 n)  (2 n + 1)
                                  /

                     2
    (2 n + 3) (n + 3)  (n + 2) (4 n + 9) (3 n + 5) (3 n + 8) (4 + 3 n)

    (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/576*(285383*n^6+3343155*n^5+16147483*n^4+41164797*n^3+58445046*n^2+43857720*
n+13608000)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(7+2*
n)/(5+2*n)^2/(2*n+1)/(2*n+3)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3
*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (5 n + 4) (5 n + 3) (5 n + 2)

                       6            5            4             3             2
    (5 n + 1) (125659 n  + 1525627 n  + 7667663 n  + 20423501 n  + 30415926 n

                               /                         2
     + 24021720 n + 7862400)  /  (576 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3)
                             /

           2
    (n + 3)  (n + 2) (4 n + 9) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/576*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(125659*
n^6+1525627*n^5+7667663*n^4+20423501*n^3+30415926*n^2+24021720*n+7862400)/(7+2*
n)/(5+2*n)^2/(2*n+1)/(2*n+3)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3
*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (5 n + 4) (5 n + 3) (5 n + 2)

                      6            5            4             3             2
    (5 n + 1) (78089 n  + 1001017 n  + 5321853 n  + 15013631 n  + 23691946 n

                               /                         2
     + 19814040 n + 6854400)  /  (192 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3)
                             /

           2
    (n + 3)  (n + 2) (4 n + 9) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/192*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(78089*n
^6+1001017*n^5+5321853*n^4+15013631*n^3+23691946*n^2+19814040*n+6854400)/(7+2*n
)/(5+2*n)^2/(2*n+1)/(2*n+3)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*
n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (5 n + 1) (5 n + 11) (5 n + 12)

    (5 n + 4) (5 n + 3) (5 n + 2)

           4          3          2                      /
    (2743 n  + 23226 n  + 72785 n  + 99822 n + 50400)  /  (576 (7 + 2 n)
                                                      /

             2                            2
    (5 + 2 n)  (2 n + 1) (2 n + 3) (n + 3)  (n + 2) (4 n + 9) (3 n + 5)

    (3 n + 8) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



125/576*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(5*n+1)*(5*n+11)*(5*n+12)*(5*n+4)*(5*n+
3)*(5*n+2)*(2743*n^4+23226*n^3+72785*n^2+99822*n+50400)/(7+2*n)/(5+2*n)^2/(2*n+
1)/(2*n+3)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                 7            6             5              4
    (5 n + 2) (5 n + 1) (493478 n  + 7210124 n  + 44772795 n  + 153181160 n

                  3              2                             /
     + 311913457 n  + 378171986 n  + 252953640 n + 72072000)  /  (192
                                                             /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9)

                   2                                         2
    (n + 2) (n + 3)  (4 n + 11) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



25/192*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
493478*n^7+7210124*n^6+44772795*n^5+153181160*n^4+311913457*n^3+378171986*n^2+
252953640*n+72072000)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)
/(n+2)/(n+3)^2/(4*n+11)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4               3
25 (1745489 n  + 26301073 n  + 168928251 n  + 599592593 n  + 1270350100 n

                   2
     + 1606846494 n  + 1123675920 n + 335134800) (5 n + 6) (5 n + 7) (5 n + 8)

                                                        /
    (5 n + 9) (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2)  /  (288 (7 + 2 n)
                                                      /

    (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (2 n + 1)

             2                                       2
    (5 + 2 n)  (4 n + 13) (4 n + 9) (2 n + 3) (n + 3) )



                             and in Maple notation



25/288*(1745489*n^7+26301073*n^6+168928251*n^5+599592593*n^4+1270350100*n^3+
1606846494*n^2+1123675920*n+335134800)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)*(5*n+1)*
(5*n+4)*(5*n+3)*(5*n+2)/(7+2*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+
8)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(2*n+3)/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6           5            4             3             2
125 (57914 n  + 774576 n  + 4293587 n  + 12621198 n  + 20738627 n  + 18044754 n

     + 6486480) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1)

                                              /
    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (576 (3 n + 10) (7 + 3 n)
                                            /

                                                                      2
    (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 3)

                                            2
    (4 n + 11) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



125/576*(57914*n^6+774576*n^5+4293587*n^4+12621198*n^3+20738627*n^2+18044754*n+
6486480)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n
+8)*(5*n+9)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+
3)^2/(4*n+11)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6           5            4            3            2
125 (16967 n  + 234920 n  + 1346682 n  + 4087838 n  + 6922699 n  + 6192950 n

     + 2282280) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1)

                                              /
    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (96 (3 n + 10) (7 + 3 n)
                                            /

                                                                      2
    (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 3)

                                            2
    (4 n + 11) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



125/96*(16967*n^6+234920*n^5+1346682*n^4+4087838*n^3+6922699*n^2+6192950*n+
2282280)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n
+8)*(5*n+9)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+
3)^2/(4*n+11)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 3, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



            5           4            3            2
125 (29138 n  + 333134 n  + 1507171 n  + 3367927 n  + 3710142 n + 1607760)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1)

                                              /
    (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (576 (3 n + 10) (7 + 3 n)
                                            /

                                                                      2
    (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 3)

                                            2
    (4 n + 11) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



125/576*(29138*n^5+333134*n^4+1507171*n^3+3367927*n^2+3710142*n+1607760)*(5*n+2
)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5
*n+9)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+3)^2/(
4*n+11)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5            4            3            2
25 (34814 n  + 413520 n  + 2009349 n  + 5104076 n  + 7139053 n  + 5211220 n

     + 1552320) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)/(18

    (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8)

    (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/18*(34814*n^6+413520*n^5+2009349*n^4+5104076*n^3+7139053*n^2+5211220*n+
1552320)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/
(7+2*n)/(n+2)/(n+3)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5           4            3            2
25 (30217 n  + 294630 n  + 1119175 n  + 2071362 n  + 1873384 n + 665280)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(24

    (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8)

    (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/24*(30217*n^5+294630*n^4+1119175*n^3+2071362*n^2+1873384*n+665280)*(5*n+1)*(
5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n
+2)/(n+3)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5          4           3           2
175 (4847 n  + 49101 n  + 195143 n  + 380739 n  + 365570 n + 138600) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)/(72 (2 n + 1)

    (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (3 n + 5) (3 n + 8)

    (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



175/72*(4847*n^5+49101*n^4+195143*n^3+380739*n^2+365570*n+138600)*(5*n+2)*(5*n+
3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2
)/(n+3)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5           4           3           2
25 (11038 n  + 111392 n  + 437585 n  + 838480 n  + 787101 n + 291060) (5 n + 1)

                                                                          /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (18
                                                                        /

                                                   2
    (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 3)  (n + 2) (3 n + 5)

    (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/18*(11038*n^5+111392*n^4+437585*n^3+838480*n^2+787101*n+291060)*(5*n+1)*(5*n
+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*
n)/(n+3)^2/(n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          5          4           3           2
25 (6277 n  + 65921 n  + 271469 n  + 549183 n  + 547358 n + 215600) (5 n + 6)

                                                                            /
    (5 n + 7) (5 n + 8) (5 n + 9) (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2)  /  (
                                                                          /

                                                      2
    24 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 3)  (n + 2) (3 n + 5)

    (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/24*(6277*n^5+65921*n^4+271469*n^3+549183*n^2+547358*n+215600)*(5*n+6)*(5*n+7
)*(5*n+8)*(5*n+9)*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*
n)/(n+3)^2/(n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          5          4           3           2
25 (8146 n  + 90824 n  + 399139 n  + 864424 n  + 922603 n + 388080) (5 n + 1)

                                                                            /
    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (
                                                                          /

                                                      2
    18 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 3)  (n + 2) (3 n + 5)

    (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/18*(8146*n^5+90824*n^4+399139*n^3+864424*n^2+922603*n+388080)*(5*n+1)*(5*n+2
)*(5*n+3)*(5*n+4)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*
n)/(n+3)^2/(n+2)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                 7            6             5              4
    (5 n + 2) (5 n + 1) (461620 n  + 7025862 n  + 45278001 n  + 160168255 n

                  3              2                             /
     + 335983385 n  + 418191593 n  + 286256124 n + 83243160)  /  (36 (3 n + 10)
                                                             /

    (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9)

                   2
    (n + 2) (n + 3)  (4 n + 11) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1))



                             and in Maple notation



25/36*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
461620*n^7+7025862*n^6+45278001*n^5+160168255*n^4+335983385*n^3+418191593*n^2+
286256124*n+83243160)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13
)/(4*n+9)/(n+2)/(n+3)^2/(4*n+11)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4               3
25 (2128424 n  + 33317716 n  + 221554522 n  + 811456665 n  + 1768397991 n

                   2
     + 2293986419 n  + 1640846823 n + 499458960) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /            2
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (72 (n + 3)
                                                      /

    (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10)

    (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



25/72*(2128424*n^7+33317716*n^6+221554522*n^5+811456665*n^4+1768397991*n^3+
2293986419*n^2+1640846823*n+499458960)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*
(5*n+7)*(5*n+8)*(5*n+9)/(n+3)^2/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)/(2*n+1)/(3*n+8
)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5             4              3
125 (168552 n  + 2750380 n  + 19108382 n  + 73260299 n  + 167362444 n

                  2
     + 227750121 n  + 170866974 n + 54486432) (5 n + 2) (5 n + 3) (5 n + 4)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (72
                                                                 /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13)

                             2
    (4 n + 9) (n + 2) (n + 3)  (4 n + 11) (7 + 2 n) (5 + 2 n) (2 n + 3)

    (2 n + 1))



                             and in Maple notation



125/72*(168552*n^7+2750380*n^6+19108382*n^5+73260299*n^4+167362444*n^3+
227750121*n^2+170866974*n+54486432)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5
*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)
/(4*n+13)/(4*n+9)/(n+2)/(n+3)^2/(4*n+11)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5          4           3           2
125 (5300 n  + 62042 n  + 286463 n  + 651433 n  + 728648 n + 320166) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 6)

                                    /
    (5 n + 7) (5 n + 8) (5 n + 9)  /  (36 (3 n + 10) (7 + 3 n) (4 + 3 n)
                                  /

                                                                       2
    (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 3)

    (4 n + 11) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1))



                             and in Maple notation



125/36*(5300*n^5+62042*n^4+286463*n^3+651433*n^2+728648*n+320166)*(5*n+2)*(5*n+
3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(
3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+3)^2
/(4*n+11)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   8              7               6
    (5 n + 2) (5 n + 1) (11762208 n  + 209986232 n  + 1626888249 n

                   5                4                3                2
     + 7144763690 n  + 19456060542 n  + 33648797558 n  + 36108666561 n

                                     /
     + 21994822080 n + 5827021200)  /  (144 (3 n + 10) (7 + 3 n) (4 + 3 n)
                                   /

                                                               2
    (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 3)  (2 n + 3)

                       2          2
    (2 n + 1) (7 + 2 n)  (5 + 2 n)  (4 n + 11))



                             and in Maple notation



25/144*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
11762208*n^8+209986232*n^7+1626888249*n^6+7144763690*n^5+19456060542*n^4+
33648797558*n^3+36108666561*n^2+21994822080*n+5827021200)/(3*n+10)/(7+3*n)/(4+3
*n)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+3)^2/(2*n+3)/(2*n+1)/(7+2*n)^2
/(5+2*n)^2/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6              5               4
125 (1151584 n  + 21084696 n  + 167915697 n  + 759791964 n  + 2136727464 n

                   3               2
     + 3824872362 n  + 4256618651 n  + 2692993254 n + 741620880) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8)

                /
    (5 n + 9)  /  (144 (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8)
              /

                                          2                              2
    (3 n + 5) (4 n + 13) (4 n + 9) (n + 3)  (2 n + 3) (2 n + 1) (7 + 2 n)

             2
    (5 + 2 n)  (4 n + 11))



                             and in Maple notation



125/144*(1151584*n^8+21084696*n^7+167915697*n^6+759791964*n^5+2136727464*n^4+
3824872362*n^3+4256618651*n^2+2692993254*n+741620880)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+
11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+3)^2/(2*n+3)/(2*n+1)/(7+2*n)^2/(5+2*n)^
2/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5             4              3
125 (100000 n  + 1639256 n  + 11450481 n  + 44176025 n  + 101640555 n

                  2
     + 139416689 n  + 105506534 n + 33957000) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8)

                /                     2
    (5 n + 9)  /  (144 (n + 2) (n + 3)  (2 n + 3) (4 n + 9) (4 n + 13)
              /

             2
    (5 + 2 n)  (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11)

                       2
    (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/144*(100000*n^7+1639256*n^6+11450481*n^5+44176025*n^4+101640555*n^3+
139416689*n^2+105506534*n+33957000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(
5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(n+2)/(n+3)^2/(2*n+3)/(4*n+9)/(
4*n+13)/(5+2*n)^2/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+
2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5             4              3
125 (184800 n  + 3140728 n  + 22731921 n  + 90785281 n  + 215941311 n

                  2
     + 305690173 n  + 238253190 n + 78780240) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8)

                /                     2
    (5 n + 9)  /  (144 (n + 2) (n + 3)  (2 n + 3) (4 n + 9) (4 n + 13)
              /

             2
    (5 + 2 n)  (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11)

                       2
    (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/144*(184800*n^7+3140728*n^6+22731921*n^5+90785281*n^4+215941311*n^3+
305690173*n^2+238253190*n+78780240)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(
5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(n+2)/(n+3)^2/(2*n+3)/(4*n+9)/(
4*n+13)/(5+2*n)^2/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+
2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 4, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6           5            4             3             2
125 (53920 n  + 777464 n  + 4625773 n  + 14526743 n  + 25373533 n  + 23346839 n

     + 8828820) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 14)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (144
                                                                 /

                   2                                         2
    (n + 2) (n + 3)  (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)  (2 n + 1)

                                                                           2
    (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n)

    (3 n + 11))



                             and in Maple notation



125/144*(53920*n^6+777464*n^5+4625773*n^4+14526743*n^3+25373533*n^2+23346839*n+
8828820)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+14)*(5*n+11)*(5*n+1)*(5
*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(n+2)/(n+3)^2/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)^2/
(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6           5            4            3             2
125 (50293 n  + 611862 n  + 3048907 n  + 7954026 n  + 11448400 n  + 8623392 n

     + 2661120) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)

    (5 n + 7)/(216 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3)

    (n + 4) (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



125/216*(50293*n^6+611862*n^5+3048907*n^4+7954026*n^3+11448400*n^2+8623392*n+
2661120)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+1)/(2*n+3)/
(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n
+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5          4           3           2
175 (4325 n  + 43356 n  + 169439 n  + 323016 n  + 301544 n + 110880) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)/(36

    (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (n + 4) (3 n + 5)

    (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



175/36*(4325*n^5+43356*n^4+169439*n^3+323016*n^2+301544*n+110880)*(5*n+1)*(5*n+
2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n
)/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 2) (5 n + 3) (5 n + 4)

            5           4           3            2
    (17846 n  + 185958 n  + 760469 n  + 1527537 n  + 1511420 n + 591360)

    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)/(54 (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (n + 4) (3 n + 5) (3 n + 8) (3 n + 11)

    (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/54*(5*n+2)*(5*n+3)*(5*n+4)*(17846*n^5+185958*n^4+760469*n^3+1527537*n^2+
1511420*n+591360)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*
n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



175 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

    (5 n + 2) (5 n + 1)

           5          4           3           2                        /
    (7409 n  + 76462 n  + 307447 n  + 603890 n  + 582312 n + 221760)  /  (108
                                                                     /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 2)

           2
    (n + 3)  (n + 4) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1))



                             and in Maple notation



175/108*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
7409*n^5+76462*n^4+307447*n^3+603890*n^2+582312*n+221760)/(3*n+10)/(7+3*n)/(4+3
*n)/(3*n+11)/(3*n+8)/(3*n+5)/(n+2)/(n+3)^2/(n+4)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1
)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5           4            3            2
25 (77201 n  + 829411 n  + 3496003 n  + 7244669 n  + 7403916 n + 2993760)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /            2
    (5 n + 6)  /  (72 (n + 3)  (n + 4) (2 n + 3) (5 + 2 n) (2 n + 1) (3 n + 8)
              /

    (3 n + 10) (7 + 3 n) (3 n + 5) (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



25/72*(77201*n^5+829411*n^4+3496003*n^3+7244669*n^2+7403916*n+2993760)*(5*n+2)*
(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+3)^2/(n+4)/(2*n+3)/(
5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            5           4           3            2
125 (17069 n  + 194434 n  + 873067 n  + 1932446 n  + 2108664 n + 907200)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /
    (5 n + 7) (5 n + 6)  /  (216 (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11)
                        /

                                       2
    (3 n + 8) (3 n + 5) (n + 2) (n + 3)  (n + 4) (7 + 2 n) (5 + 2 n) (2 n + 3)

    (2 n + 1))



                             and in Maple notation



125/216*(17069*n^5+194434*n^4+873067*n^3+1932446*n^2+2108664*n+907200)*(5*n+2)*
(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+10)/(7+3*
n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(n+2)/(n+3)^2/(n+4)/(7+2*n)/(5+2*n)/(2*n+3)
/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   7              6               5
    (5 n + 2) (5 n + 1) (30130172 n  + 463338144 n  + 3019850729 n

                    4                3                2
     + 10815972240 n  + 23001935123 n  + 29068114776 n  + 20234085696 n

                     /
     + 5993507520)  /  (432 (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8)
                   /

                                          2
    (3 n + 5) (4 n + 13) (4 n + 9) (n + 3)  (n + 4) (7 + 2 n) (5 + 2 n)

    (2 n + 3) (2 n + 1) (4 n + 11))



                             and in Maple notation



25/432*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
30130172*n^7+463338144*n^6+3019850729*n^5+10815972240*n^4+23001935123*n^3+
29068114776*n^2+20234085696*n+5993507520)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*
n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+3)^2/(n+4)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(4*
n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              7             6              5               4               3
125 (2901836 n  + 45916896 n  + 308876597 n  + 1145332014 n  + 2529219329 n

                   2
     + 3327594804 n  + 2416197888 n + 747242496) (5 n + 2) (5 n + 3) (5 n + 4)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (432
                                                                 /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13)

                     2
    (4 n + 9) (n + 3)  (n + 4) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1)

    (4 n + 11))



                             and in Maple notation



125/432*(2901836*n^7+45916896*n^6+308876597*n^5+1145332014*n^4+2529219329*n^3+
3327594804*n^2+2416197888*n+747242496)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)
*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n
+5)/(4*n+13)/(4*n+9)/(n+3)^2/(n+4)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5              4              3
125 (406532 n  + 6703024 n  + 47077199 n  + 182542060 n  + 421952303 n

                  2
     + 581272006 n  + 441666816 n + 142702560) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)

       /                     2
      /  (144 (n + 2) (n + 3)  (n + 4) (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)
     /

    (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n)

    (7 + 2 n) (3 n + 11))



                             and in Maple notation



125/144*(406532*n^7+6703024*n^6+47077199*n^5+182542060*n^4+421952303*n^3+
581272006*n^2+441666816*n+142702560)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*
(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)/(2*n+3)/(4*n+9)/(4*
n+13)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)
/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



            5            4            3             2
125 (83612 n  + 1008396 n  + 4798805 n  + 11246175 n  + 12954458 n + 5854464)

                                            2
    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12)  (5 n + 13) (5 n + 11) (5 n + 1)

                                              /                     2
    (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (432 (n + 2) (n + 3)  (n + 4)
                                            /

    (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10)

    (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



125/432*(83612*n^5+1008396*n^4+4798805*n^3+11246175*n^2+12954458*n+5854464)*(5*
n+2)*(5*n+3)*(5*n+4)*(5*n+12)^2*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+
7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)/(2*n+1)/(3*n+8)
/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4)

                                            7             6              5
    (5 n + 3) (5 n + 2) (5 n + 1) (2518412 n  + 39033744 n  + 256724045 n

                  4               3               2
     + 929115546 n  + 1999540481 n  + 2561042040 n  + 1809610272 n + 544864320)

       /
      /  (144 (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5)
     /

                                2
    (4 n + 13) (4 n + 9) (n + 3)  (n + 4) (7 + 2 n) (5 + 2 n) (2 n + 3)

    (2 n + 1) (4 n + 15) (4 n + 11))



                             and in Maple notation



125/144*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*
(5*n+1)*(2518412*n^7+39033744*n^6+256724045*n^5+929115546*n^4+1999540481*n^3+
2561042040*n^2+1809610272*n+544864320)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8
)/(3*n+5)/(4*n+13)/(4*n+9)/(n+3)^2/(n+4)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(4*n+
15)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              7             6              5               4               3
125 (3832516 n  + 61172280 n  + 415493167 n  + 1557215556 n  + 3479247787 n

                   2
     + 4635888654 n  + 3412001340 n + 1070269200) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)

       /
      /  (432 (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5)
     /

                                2
    (4 n + 13) (4 n + 9) (n + 3)  (n + 4) (7 + 2 n) (5 + 2 n) (2 n + 3)

    (2 n + 1) (4 n + 15) (4 n + 11))



                             and in Maple notation



125/432*(3832516*n^7+61172280*n^6+415493167*n^5+1557215556*n^4+3479247787*n^3+
4635888654*n^2+3412001340*n+1070269200)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+
11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(
3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+3)^2/(n+4)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(
4*n+15)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5              4              3
625 (348436 n  + 5796768 n  + 41104411 n  + 161015754 n  + 376219009 n

                  2
     + 524130654 n  + 402901128 n + 131725440) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /                     2
    (5 n + 6)  /  (432 (n + 2) (n + 3)  (n + 4) (2 n + 3) (4 n + 9) (4 n + 13)
              /

    (5 + 2 n) (2 n + 1) (4 n + 15) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5)

    (4 n + 11) (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



625/432*(348436*n^7+5796768*n^6+41104411*n^5+161015754*n^4+376219009*n^3+
524130654*n^2+402901128*n+131725440)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*
(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)/(2*n+3)/(4
*n+9)/(4*n+13)/(5+2*n)/(2*n+1)/(4*n+15)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+
11)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 2) (5 n + 3) (5 n + 4)

            5           4            3            2
    (25292 n  + 302004 n  + 1423761 n  + 3308229 n  + 3782404 n + 1698840)

    (5 n + 12) (5 n + 13) (5 n + 14) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                    2
    (5 n + 7) (5 n + 6)  /  (48 (n + 2) (n + 3)  (n + 4) (2 n + 3) (4 n + 9)
                        /

    (4 n + 13) (5 + 2 n) (2 n + 1) (4 n + 15) (3 n + 10) (7 + 3 n) (3 n + 5)

    (4 n + 11) (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



125/48*(5*n+2)*(5*n+3)*(5*n+4)*(25292*n^5+302004*n^4+1423761*n^3+3308229*n^2+
3782404*n+1698840)*(5*n+12)*(5*n+13)*(5*n+14)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*
(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)/(2*n+1)/(4
*n+15)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, 1 + n, n + 5, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                                             6
125 (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 14) (1068212 n

                 5             4              3              2
     + 15976576 n  + 98522755 n  + 320327810 n  + 578474883 n  + 549411984 n

     + 214053840) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)

       /                     2
      /  (432 (n + 2) (n + 3)  (n + 4) (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)
     /

    (2 n + 1) (4 n + 15) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11)

    (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



125/432*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+14)*(1068212*n^6+
15976576*n^5+98522755*n^4+320327810*n^3+578474883*n^2+549411984*n+214053840)*(5
*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)/(2*n+3)/(4*n
+9)/(4*n+13)/(5+2*n)/(2*n+1)/(4*n+15)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)
/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                   7           6            5
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (39984 n  + 379212 n  + 1443696 n

                4            3            2                       /
     + 2849155 n  + 3128419 n  + 1897934 n  + 586040 n + 70560)  /  (64
                                                                /

                                           2                              2
    (7 + 2 n) (4 n + 3) (5 + 4 n) (2 n + 1)  (5 + 2 n) (4 n + 7) (2 n + 3)

    (4 n + 1) (n + 4))



                             and in Maple notation



15/64*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(39984*n^7+379212*n^6+1443696*n^5+2849155
*n^4+3128419*n^3+1897934*n^2+586040*n+70560)/(7+2*n)/(4*n+3)/(5+4*n)/(2*n+1)^2/
(5+2*n)/(4*n+7)/(2*n+3)^2/(4*n+1)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                   8            7             6
5 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (630368 n  + 6873312 n  + 31145340 n

                 5              4             3             2
     + 76547154 n  + 111500349 n  + 98495031 n  + 51497542 n  + 14548464 n

                  /                                             2
     + 1693440)  /  (128 (7 + 2 n) (4 n + 3) (5 + 4 n) (2 n + 1)  (5 + 2 n)
                /

                       2
    (4 n + 7) (2 n + 3)  (4 n + 1) (n + 4) (n + 2))



                             and in Maple notation



5/128*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(630368*n^8+6873312*n^7+31145340*n^6+
76547154*n^5+111500349*n^4+98495031*n^3+51497542*n^2+14548464*n+1693440)/(7+2*n
)/(4*n+3)/(5+4*n)/(2*n+1)^2/(5+2*n)/(4*n+7)/(2*n+3)^2/(4*n+1)/(n+4)/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                    8            7
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (336288 n  + 3872928 n

                 6             5             4             3             2
     + 18741140 n  + 49774814 n  + 79308647 n  + 77515637 n  + 45262370 n

                               /                                             2
     + 14366576 n + 1881600)  /  (128 (7 + 2 n) (4 n + 3) (5 + 4 n) (2 n + 1)
                             /

                                 2
    (5 + 2 n) (4 n + 7) (2 n + 3)  (4 n + 1) (n + 4) (n + 2))



                             and in Maple notation



15/128*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(336288*n^8+3872928*n^7+18741140*n^6+
49774814*n^5+79308647*n^4+77515637*n^3+45262370*n^2+14366576*n+1881600)/(7+2*n)
/(4*n+3)/(5+4*n)/(2*n+1)^2/(5+2*n)/(4*n+7)/(2*n+3)^2/(4*n+1)/(n+4)/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            8             7              6              5               4
5 (5746712 n  + 65733082 n  + 313691165 n  + 815067916 n  + 1260372863 n

                   3              2
     + 1187304718 n  + 665065560 n  + 202152384 n + 25401600) (5 n + 1)

                                            /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)  /  (128 (n + 4) (n + 2) (5 + 2 n)
                                          /

                       2          2
    (7 + 2 n) (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3)

    (4 n + 7))



                             and in Maple notation



5/128*(5746712*n^8+65733082*n^7+313691165*n^6+815067916*n^5+1260372863*n^4+
1187304718*n^3+665065560*n^2+202152384*n+25401600)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n
+4)*(1+n)/(n+4)/(n+2)/(5+2*n)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+
9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            8            7            6             5             4
75 (145064 n  + 1752678 n  + 8925903 n  + 25017510 n  + 42170061 n

                 3             2
     + 43698117 n  + 27095827 n  + 9138360 n + 1270080) (5 n + 2) (5 n + 3)

                                    /
    (5 n + 4) (5 n + 1) (5 n + 6)  /  (128 (n + 4) (n + 2) (5 + 2 n) (7 + 2 n)
                                  /

             2          2
    (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



75/128*(145064*n^8+1752678*n^7+8925903*n^6+25017510*n^5+42170061*n^4+43698117*n
^3+27095827*n^2+9138360*n+1270080)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)/(n+4
)/(n+2)/(5+2*n)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*
n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                     8             7
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (2836452 n  + 34818685 n

                  6              5              4              3              2
     + 180542681 n  + 516293017 n  + 889640153 n  + 943922560 n  + 600049124 n

                                 /
     + 207664128 n + 29635200)  /  (128 (7 + 2 n) (4 n + 3) (5 + 4 n)
                               /

             2
    (2 n + 1)  (5 + 2 n) (4 n + 9) (4 n + 7) (2 n + 3) (4 n + 1) (n + 4)

    (n + 2))



                             and in Maple notation



15/128*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(2836452*n^8+34818685*n^7+180542681*n^6+
516293017*n^5+889640153*n^4+943922560*n^3+600049124*n^2+207664128*n+29635200)/(
7+2*n)/(4*n+3)/(5+4*n)/(2*n+1)^2/(5+2*n)/(4*n+9)/(4*n+7)/(2*n+3)/(4*n+1)/(n+4)/
(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            8            7             6              5              4
75 (641264 n  + 7634764 n  + 38029840 n  + 103439157 n  + 167971741 n

                  3             2
     + 166723416 n  + 98706410 n  + 31783808 n + 4233600) (5 n + 1) (5 n + 2)

                                            /
    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)  /  (256 (n + 4) (n + 2) (7 + 2 n)
                                          /

             2          2          2
    (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3)

    (4 n + 7))



                             and in Maple notation



75/256*(641264*n^8+7634764*n^7+38029840*n^6+103439157*n^5+167971741*n^4+
166723416*n^3+98706410*n^2+31783808*n+4233600)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*
(1+n)*(5*n+6)/(n+4)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)
/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            8            7             6             5             4
75 (270064 n  + 3395020 n  + 18027952 n  + 52786189 n  + 93116341 n

                  3             2
     + 101131150 n  + 65788668 n  + 23276736 n + 3386880) (5 n + 2) (5 n + 3)

                                              /
    (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (256 (n + 4) (n + 2) (7 + 2 n)
                                            /

             2          2          2
    (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3)

    (4 n + 7))



                             and in Maple notation



75/256*(270064*n^8+3395020*n^7+18027952*n^6+52786189*n^5+93116341*n^4+101131150
*n^3+65788668*n^2+23276736*n+3386880)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(
5*n+6)/(n+4)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9
)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                      10               9
5 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (186466256 n   + 2956915796 n

                    8                7                 6                 5
     + 20658081900 n  + 83728026423 n  + 217927143624 n  + 380282458719 n

                     4                 3                 2
     + 449901202210 n  + 355524290992 n  + 178979153280 n  + 51561417600 n

                     /                                         2          2
     + 6401203200)  /  (256 (n + 4) (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)
                   /

             2
    (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



5/256*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(186466256*n^10+2956915796*n^9+
20658081900*n^8+83728026423*n^7+217927143624*n^6+380282458719*n^5+449901202210*
n^4+355524290992*n^3+178979153280*n^2+51561417600*n+6401203200)/(n+4)/(n+2)/(7+
2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                      10               9
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (95590832 n   + 1554897132 n

                    8                7                 6                 5
     + 11153758700 n  + 46447388121 n  + 124257246978 n  + 222880415313 n

                     4                 3                 2
     + 270989453320 n  + 219973141284 n  + 113668442560 n  + 33576012960 n

                     /                                         2          2
     + 4267468800)  /  (256 (n + 4) (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)
                   /

             2
    (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



15/256*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(95590832*n^10+1554897132*n^9+
11153758700*n^8+46447388121*n^7+124257246978*n^6+222880415313*n^5+270989453320*
n^4+219973141284*n^3+113668442560*n^2+33576012960*n+4267468800)/(n+4)/(n+2)/(7+
2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (6070976 n  + 74787528 n  + 386275890 n  + 1091936793 n  + 1847475375 n

                   3               2
     + 1915455486 n  + 1187097976 n  + 400611456 n + 55883520) (5 n + 1)

                                                                /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (512 (n + 4)
                                                              /

                               2          2          2
    (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/512*(6070976*n^8+74787528*n^7+386275890*n^6+1091936793*n^5+1847475375*n^4+
1915455486*n^3+1187097976*n^2+400611456*n+55883520)*(5*n+1)*(5*n+2)*(5*n+3)*(5*
n+4)*(1+n)*(5*n+6)*(5*n+7)/(n+4)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4
*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6              5               4
25 (2759936 n  + 35878536 n  + 197312430 n  + 599181393 n  + 1097651094 n

                   3              2
     + 1239352149 n  + 838658830 n  + 308536032 n + 46569600) (5 n + 2)

                                                                  /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)  /  (512
                                                                /

                                       2          2          2
    (n + 4) (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1)

    (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/512*(2759936*n^8+35878536*n^7+197312430*n^6+599181393*n^5+1097651094*n^4+
1239352149*n^3+838658830*n^2+308536032*n+46569600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n
+1)*(5*n+8)*(5*n+7)*(5*n+6)/(n+4)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(
4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        11                10
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (1174789056 n   + 21235330808 n

                     9                 8                  7                  6
     + 171548098934 n  + 817436052095 n  + 2551824577725 n  + 5476011172293 n

                      5                  4                  3
     + 8233773624321 n  + 8660932906420 n  + 6231487920604 n

                      2                                   /
     + 2911023072064 n  + 790713262080 n + 93884313600)  /  (512 (n + 4)
                                                        /

                               2          2          2
    (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



15/512*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(1174789056*n^11+21235330808*n^10+
171548098934*n^9+817436052095*n^8+2551824577725*n^7+5476011172293*n^6+
8233773624321*n^5+8660932906420*n^4+6231487920604*n^3+2911023072064*n^2+
790713262080*n+93884313600)/(n+4)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(
4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                       11                 10
5 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (6353274496 n   + 118483154376 n

                     9                  8                   7
     + 988075923614 n  + 4861566591387 n  + 15670142275839 n

                       6                   5                   4
     + 34709257413081 n  + 53837160282371 n  + 58368984584088 n

                       3                   2
     + 43237907242540 n  + 20767188996768 n  + 5789968309440 n + 704132352000)

       /                                         2          2          2
      /  (512 (n + 4) (n + 2) (7 + 2 n) (2 n + 1)  (2 n + 3)  (5 + 2 n)
     /

    (4 n + 1) (5 + 4 n) (4 n + 9) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



5/512*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(6353274496*n^11+118483154376*n^10+
988075923614*n^9+4861566591387*n^8+15670142275839*n^7+34709257413081*n^6+
53837160282371*n^5+58368984584088*n^4+43237907242540*n^3+20767188996768*n^2+
5789968309440*n+704132352000)/(n+4)/(n+2)/(7+2*n)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2
/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        10                9
25 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (1928238272 n   + 32769324296 n

                     8                  7                  6                  5
     + 245596257954 n  + 1068224567361 n  + 2983088954385 n  + 5580797131629 n

                      4                  3                  2
     + 7069644403141 n  + 5971459774594 n  + 3205881798288 n  + 981977834880 n

                       /
     + 129090931200)  /  (512 (7 + 2 n) (4 n + 11) (4 n + 3) (5 + 4 n)
                     /

             2          2                              2
    (2 n + 1)  (5 + 2 n)  (4 n + 9) (4 n + 7) (2 n + 3)  (4 n + 1) (n + 4))



                             and in Maple notation



25/512*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(1928238272*n^10+32769324296*n^9+
245596257954*n^8+1068224567361*n^7+2983088954385*n^6+5580797131629*n^5+
7069644403141*n^4+5971459774594*n^3+3205881798288*n^2+981977834880*n+
129090931200)/(7+2*n)/(4*n+11)/(4*n+3)/(5+4*n)/(2*n+1)^2/(5+2*n)^2/(4*n+9)/(4*n
+7)/(2*n+3)^2/(4*n+1)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                3         2
5 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (173 n  + 1065 n  + 1954 n + 1008)/(

    16 (7 + 2 n) (2 n + 1) (5 + 2 n) (2 n + 3) (n + 2) (n + 3) (n + 4))



                             and in Maple notation



5/16*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(173*n^3+1065*n^2+1954*n+1008)/(7+2*n)/(2*
n+1)/(5+2*n)/(2*n+3)/(n+2)/(n+3)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6            5            4             3             2
5 (102253 n  + 1126415 n  + 5009441 n  + 11539333 n  + 14549226 n  + 9534672 n

                                                                  /
     + 2540160) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)  /  (32
                                                                /

                                 2
    (5 + 2 n) (7 + 2 n) (2 n + 3)  (2 n + 1) (n + 2) (n + 3) (n + 4) (5 + 4 n)

    (4 n + 9) (4 n + 7))



                             and in Maple notation



5/32*(102253*n^6+1126415*n^5+5009441*n^4+11539333*n^3+14549226*n^2+9534672*n+
2540160)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(5+2*n)/(7+2*n)/(2*n+3)^2/(2*n+1
)/(n+2)/(n+3)/(n+4)/(5+4*n)/(4*n+9)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          5          4          3           2
25 (2578 n  + 25654 n  + 98987 n  + 185621 n  + 169560 n + 60480) (5 n + 2)

                                                        /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (64 (5 + 2 n)
                                                      /

                       2
    (7 + 2 n) (2 n + 3)  (2 n + 1) (n + 2) (n + 3) (n + 4) (5 + 4 n) (4 n + 9)

    (4 n + 7))



                             and in Maple notation



25/64*(2578*n^5+25654*n^4+98987*n^3+185621*n^2+169560*n+60480)*(5*n+2)*(5*n+3)*
(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(5+2*n)/(7+2*n)/(2*n+3)^2/(2*n+1)/(n+2)/(n+3)/(
n+4)/(5+4*n)/(4*n+9)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          5          4           3           2
75 (4488 n  + 44666 n  + 171673 n  + 319069 n  + 287504 n + 100800) (5 n + 1)

                                                                /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (128
                                                              /

                       2          2
    (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (n + 4) (4 n + 7)

    (5 + 4 n) (4 n + 9))



                             and in Maple notation



75/128*(4488*n^5+44666*n^4+171673*n^3+319069*n^2+287504*n+100800)*(5*n+1)*(5*n+
2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n
+2)/(n+3)/(n+4)/(4*n+7)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          6          5           4            3            2
75 (8456 n  + 99474 n  + 476915 n  + 1196010 n  + 1658969 n  + 1209576 n

                                                                             /
     + 362880) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /
                                                                           /

                            2          2
    (128 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (n + 4)

    (4 n + 7) (5 + 4 n) (4 n + 9))



                             and in Maple notation



75/128*(8456*n^6+99474*n^5+476915*n^4+1196010*n^3+1658969*n^2+1209576*n+362880)
*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2
*n+1)/(n+2)/(n+3)/(n+4)/(4*n+7)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



          4          3          2
25 (1934 n  + 17043 n  + 54526 n  + 75177 n + 37800) (5 n + 2) (5 n + 3)

                                                        /
    (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)  /  (128 (7 + 2 n)
                                                      /

                                 2                    2
    (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 9) (2 n + 3)  (n + 2) (n + 3) (n + 4)

    )



                             and in Maple notation



25/128*(1934*n^4+17043*n^3+54526*n^2+75177*n+37800)*(5*n+2)*(5*n+3)*(5*n+4)*(5*
n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+9)/(2*n+3)^
2/(n+2)/(n+3)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5            4             3             2
25 (103142 n  + 1202559 n  + 5692190 n  + 14039724 n  + 19092701 n

     + 13617504 n + 3991680) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)

                          /                         2          2
    (5 n + 6) (5 n + 7)  /  (128 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1)
                        /

    (n + 2) (n + 3) (n + 4) (5 + 4 n) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/128*(103142*n^6+1202559*n^5+5692190*n^4+14039724*n^3+19092701*n^2+13617504*n
+3991680)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(7+2*n)/(2*n+3)
^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5           4            3            2
75 (14482 n  + 175569 n  + 869802 n  + 2260021 n  + 3257126 n  + 2474840 n

     + 776160) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7)

                /                         2          2
    (5 n + 6)  /  (128 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2)
              /

    (n + 3) (n + 4) (5 + 4 n) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/128*(14482*n^6+175569*n^5+869802*n^4+2260021*n^3+3257126*n^2+2474840*n+
776160)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+3)
^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5         4          3          2
1125 (350 n  + 3831 n  + 16452 n  + 34716 n  + 36067 n + 14784) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                           2          2
    128 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (n + 4)

    (5 + 4 n) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



1125/128*(350*n^5+3831*n^4+16452*n^3+34716*n^2+36067*n+14784)*(5*n+2)*(5*n+3)*(
5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n
+1)/(n+2)/(n+3)/(n+4)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5              4              3
25 (588638 n  + 8631395 n  + 53606336 n  + 182968973 n  + 370987316 n

                  2
     + 447187502 n  + 296920560 n + 83825280) (5 n + 2) (5 n + 3) (5 n + 4)

                                    /                         2          2
    (5 n + 1) (5 n + 7) (5 n + 6)  /  (128 (7 + 2 n) (2 n + 3)  (5 + 2 n)
                                  /

    (2 n + 1) (n + 2) (n + 3) (n + 4) (5 + 4 n) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/128*(588638*n^7+8631395*n^6+53606336*n^5+182968973*n^4+370987316*n^3+
447187502*n^2+296920560*n+83825280)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*
n+6)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(5+4*n)/(4*n+9)/(4*n
+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5            4             3             2
25 (75158 n  + 897295 n  + 4358896 n  + 11061533 n  + 15522666 n  + 11463732 n

     + 3492720) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)

                          /                         2          2
    (5 n + 7) (5 n + 8)  /  (256 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1)
                        /

                   2
    (n + 4) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/256*(75158*n^6+897295*n^5+4358896*n^4+11061533*n^3+15522666*n^2+11463732*n+
3492720)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(7+2*n)/
(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+2)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11
)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5            4            3            2
25 (34262 n  + 426255 n  + 2171744 n  + 5815797 n  + 8658254 n  + 6812088 n

     + 2217600) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                         2          2
    (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1)
                        /

                   2
    (n + 4) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/256*(34262*n^6+426255*n^5+2171744*n^4+5815797*n^3+8658254*n^2+6812088*n+
2217600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n
)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+2)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+
11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5            4             3             2
25 (69358 n  + 920955 n  + 5035276 n  + 14528283 n  + 23358766 n  + 19863522 n

     + 6985440) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                         2          2
    (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n) (2 n + 3)  (5 + 2 n)  (2 n + 1)
                        /

                   2
    (n + 4) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/256*(69358*n^6+920955*n^5+5035276*n^4+14528283*n^3+23358766*n^2+19863522*n+
6985440)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n
)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+2)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+
11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (3063094 n  + 52626565 n  + 392216563 n  + 1656970399 n  + 4341830173 n

                   3               2
     + 7228753756 n  + 7470233850 n  + 4382308800 n + 1117670400) (5 n + 2)

                                                        /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n)
                                                      /

             2          2                          2
    (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 4) (n + 3)  (n + 2) (5 + 4 n)

    (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/256*(3063094*n^8+52626565*n^7+392216563*n^6+1656970399*n^5+4341830173*n^4+
7228753756*n^3+7470233850*n^2+4382308800*n+1117670400)*(5*n+2)*(5*n+3)*(5*n+4)*
(5*n+1)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+2)
/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, 1 + n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (4516174 n  + 78777289 n  + 596031991 n  + 2555665507 n  + 6794443993 n

                    3                2
     + 11472097012 n  + 12016834962 n  + 7141611312 n + 1844156160) (5 n + 2)

                                                        /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n)
                                                      /

             2          2                          2
    (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 4) (n + 3)  (n + 2) (5 + 4 n)

    (4 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/256*(4516174*n^8+78777289*n^7+596031991*n^6+2555665507*n^5+6794443993*n^4+
11472097012*n^3+12016834962*n^2+7141611312*n+1844156160)*(5*n+2)*(5*n+3)*(5*n+4
)*(5*n+1)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+
2)/(5+4*n)/(4*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           7             6             5              4              3
5 (903803 n  + 12984330 n  + 78322211 n  + 256742268 n  + 493070396 n

                  2
     + 553673952 n  + 335845440 n + 84672000) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n)/(72 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2)

    (n + 3) (n + 4) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



5/72*(903803*n^7+12984330*n^6+78322211*n^5+256742268*n^4+493070396*n^3+
553673952*n^2+335845440*n+84672000)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(2*n+
1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3
*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5           4            3            2
25 (11476 n  + 137035 n  + 663897 n  + 1668916 n  + 2294132 n  + 1633744 n

     + 470400) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)/(12

    (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (n + 4) (3 n + 5)

    (3 n + 8) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/12*(11476*n^6+137035*n^5+663897*n^4+1668916*n^3+2294132*n^2+1633744*n+470400
)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)
/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           5          4          3           2
125 (1811 n  + 19178 n  + 78937 n  + 157474 n  + 151848 n + 56448) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)/(144 (2 n + 1)

    (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (n + 4) (3 n + 5) (3 n + 8)

    (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



125/144*(1811*n^5+19178*n^4+78937*n^3+157474*n^2+151848*n+56448)*(5*n+2)*(5*n+3
)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)
/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4              3
25 (1250635 n  + 18088137 n  + 110329405 n  + 367704339 n  + 722659912 n

                  2
     + 836740164 n  + 527840208 n + 139708800) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)/(288 (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (n + 4) (4 n + 9) (3 n + 5) (3 n + 8)

    (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/288*(1250635*n^7+18088137*n^6+110329405*n^5+367704339*n^4+722659912*n^3+
836740164*n^2+527840208*n+139708800)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n
+6)*(5*n+7)*(5*n+8)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(4*n+9)/(
3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            8             7             6              5              4
25 (803225 n  + 13145404 n  + 93033114 n  + 371905618 n  + 918458637 n

                   3               2
     + 1434665002 n  + 1383799144 n  + 753166176 n + 176964480) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)/(96 (2 n + 1)

    (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (n + 4) (4 n + 9) (3 n + 5)

    (3 n + 8) (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/96*(803225*n^8+13145404*n^7+93033114*n^6+371905618*n^5+918458637*n^4+
1434665002*n^3+1383799144*n^2+753166176*n+176964480)*(5*n+2)*(5*n+3)*(5*n+4)*(5
*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)
/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7             6             5              4              3
25 (713743 n  + 10621993 n  + 66836641 n  + 230374039 n  + 469404244 n

                  2
     + 564790348 n  + 371020272 n + 102453120) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)/(288 (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (n + 2) (n + 3) (n + 4) (4 n + 9) (3 n + 5) (3 n + 8)

    (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/288*(713743*n^7+10621993*n^6+66836641*n^5+230374039*n^4+469404244*n^3+
564790348*n^2+371020272*n+102453120)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5
*n+8)*(5*n+7)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(4*n+9)
/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9              8               7               6                5
25 (8466205 n  + 158339296 n  + 1303111888 n  + 6193043362 n  + 18727158487 n

                    4                3                2
     + 37357170550 n  + 49144335420 n  + 41095565352 n  + 19810556640 n

     + 4191264000) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)

                          /                         2
    (5 n + 7) (5 n + 8)  /  (576 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3)
                        /

                   2
    (n + 4) (n + 3)  (n + 2) (4 n + 9) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/576*(8466205*n^9+158339296*n^8+1303111888*n^7+6193043362*n^6+18727158487*n^5
+37357170550*n^4+49144335420*n^3+41095565352*n^2+19810556640*n+4191264000)*(5*n
+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(7+2*n)/(5+2*n)^2/(2*
n+1)/(2*n+3)/(n+4)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+
10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7               6               5
25 (3620981 n  + 68994874 n  + 579356320 n  + 2813617534 n  + 8707579747 n

                    4                3                2
     + 17804373400 n  + 24043380512 n  + 20667612792 n  + 10254414240 n

     + 2235340800) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                         2
    (5 n + 7) (5 n + 6)  /  (576 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3)
                        /

                   2
    (n + 4) (n + 3)  (n + 2) (4 n + 9) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/576*(3620981*n^9+68994874*n^8+579356320*n^7+2813617534*n^6+8707579747*n^5+
17804373400*n^4+24043380512*n^3+20667612792*n^2+10254414240*n+2235340800)*(5*n+
2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(5+2*n)^2/(2
*n+1)/(2*n+3)/(n+4)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+
10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7               6               5
25 (2158651 n  + 42395064 n  + 367266340 n  + 1841393194 n  + 5886174937 n

                    4                3                2
     + 12433883110 n  + 17344992192 n  + 15394527232 n  + 7879663680 n

     + 1769644800) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                         2
    (5 n + 7) (5 n + 6)  /  (192 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3)
                        /

                   2
    (n + 4) (n + 3)  (n + 2) (4 n + 9) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/192*(2158651*n^9+42395064*n^8+367266340*n^7+1841393194*n^6+5886174937*n^5+
12433883110*n^4+17344992192*n^3+15394527232*n^2+7879663680*n+1769644800)*(5*n+2
)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(5+2*n)^2/(2*
n+1)/(2*n+3)/(n+4)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+
10)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8            7             6              5              4
125 (373237 n  + 6593899 n  + 50463091 n  + 218370865 n  + 583962892 n

                  3               2
     + 987238648 n  + 1029137880 n  + 603878688 n + 152409600) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /                         2
    (5 n + 6)  /  (576 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3) (n + 4)
              /

           2
    (n + 3)  (n + 2) (4 n + 9) (3 n + 5) (3 n + 8) (4 + 3 n) (7 + 3 n)

    (3 n + 10) (4 n + 7) (4 n + 11))



                             and in Maple notation



125/576*(373237*n^8+6593899*n^7+50463091*n^6+218370865*n^5+583962892*n^4+
987238648*n^3+1029137880*n^2+603878688*n+152409600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*
n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(5+2*n)^2/(2*n+1)/(2*n+3)
/(n+4)/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+7)/(
4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                  9              8               7
    (5 n + 2) (5 n + 1) (7388126 n  + 139243635 n  + 1155614223 n

                   6                5                4                3
     + 5542553161 n  + 16928107739 n  + 34136624640 n  + 45438701852 n

                    2                                 /
     + 38481714464 n  + 18804125760 n + 4036032000)  /  (384 (4 n + 11)
                                                    /

    (4 n + 7) (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13)

                             2                                      2
    (4 n + 9) (n + 2) (n + 3)  (n + 4) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n)

    )



                             and in Maple notation



25/384*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
7388126*n^9+139243635*n^8+1155614223*n^7+5542553161*n^6+16928107739*n^5+
34136624640*n^4+45438701852*n^3+38481714464*n^2+18804125760*n+4036032000)/(4*n+
11)/(4*n+7)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+
3)^2/(n+4)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7                6
25 (51132226 n  + 983735329 n  + 8345820200 n  + 40975066344 n

                     5                 4                 3                 2
     + 128276543872 n  + 265473170015 n  + 363040391942 n  + 316149260712 n

     + 158952245760 n + 35113478400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1)

                                              /
    (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1152 (7 + 2 n) (4 + 3 n)
                                            /

                                                                           2
    (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (2 n + 1) (5 + 2 n)

                                                            2
    (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3) (n + 4) (n + 3) )



                             and in Maple notation



25/1152*(51132226*n^9+983735329*n^8+8345820200*n^7+40975066344*n^6+128276543872
*n^5+265473170015*n^4+363040391942*n^3+316149260712*n^2+158952245760*n+
35113478400)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7
+2*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(2*n+1)/(5+2*n)^2/(4*n+
13)/(4*n+9)/(4*n+7)/(2*n+3)/(n+4)/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7               6                5
125 (4117594 n  + 81815537 n  + 717385739 n  + 3642028091 n  + 11792479967 n

                    4                3                2
     + 25238987972 n  + 35678603548 n  + 32091395760 n  + 16644362112 n

     + 3786531840) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9)

                                    /
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (1152 (4 n + 11) (4 n + 7) (3 n + 10)
                                  /

    (7 + 3 n) (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2)

           2                                      2
    (n + 3)  (n + 4) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



125/1152*(4117594*n^9+81815537*n^8+717385739*n^7+3642028091*n^6+11792479967*n^5
+25238987972*n^4+35678603548*n^3+32091395760*n^2+16644362112*n+3786531840)*(5*n
+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(4*n+11)/(
4*n+7)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+3)^2/
(n+4)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8            7             6              5              4
125 (473566 n  + 8481501 n  + 65813187 n  + 288784043 n  + 783040003 n

                   3               2
     + 1342032816 n  + 1417782004 n  + 842657200 n + 215255040) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /
    (5 n + 7) (5 n + 6)  /  (384 (4 n + 11) (4 n + 7) (3 n + 10) (7 + 3 n)
                        /

                                                                      2
    (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 3)  (n + 4)

                                 2
    (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



125/384*(473566*n^8+8481501*n^7+65813187*n^6+288784043*n^5+783040003*n^4+
1342032816*n^3+1417782004*n^2+842657200*n+215255040)*(5*n+2)*(5*n+3)*(5*n+4)*(5
*n+12)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(4*n+11)/(4*n+7)/(3*n+
10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+3)^2/(n+4)/(2*n+3
)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 2, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7               6
125 (10089386 n  + 206726245 n  + 1867677943 n  + 9760042663 n

                    5                4                 3                2
     + 32491171219 n  + 71400876724 n  + 103481219324 n  + 95268624528 n

     + 50484607488 n + 11711831040) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11)

                                                        /
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1152 (4 n + 11)
                                                      /

    (4 n + 7) (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 8) (3 n + 5) (4 n + 13)

                             2                                      2
    (4 n + 9) (n + 2) (n + 3)  (n + 4) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n)

    )



                             and in Maple notation



125/1152*(10089386*n^9+206726245*n^8+1867677943*n^7+9760042663*n^6+32491171219*
n^5+71400876724*n^4+103481219324*n^3+95268624528*n^2+50484607488*n+11711831040)
*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(4*n+
11)/(4*n+7)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+
3)^2/(n+4)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5             4             3
175 (113003 n  + 1654163 n  + 10176583 n  + 34058981 n  + 66866110 n

                 2
     + 76881176 n  + 47855744 n + 12418560) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6)/(144 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n)

    (n + 2) (n + 3) (n + 4) (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



175/144*(113003*n^7+1654163*n^6+10176583*n^5+34058981*n^4+66866110*n^3+76881176
*n^2+47855744*n+12418560)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(2*n+1)
/(2*n+3)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+
3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5            4             3             2
25 (145891 n  + 1791526 n  + 8924109 n  + 23064434 n  + 32608664 n

     + 23906496 n + 7096320) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)

    (5 n + 6) (5 n + 7)/(48 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2)

    (n + 3) (n + 4) (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n)

    (3 n + 10))



                             and in Maple notation



25/48*(145891*n^6+1791526*n^5+8924109*n^4+23064434*n^3+32608664*n^2+23906496*n+
7096320)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(2*n+1)/(2*n+3)/
(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n
+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5            4            3            2
25 (39245 n  + 492989 n  + 2521869 n  + 6721947 n  + 9844846 n  + 7509424 n

     + 2328480) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7)

    (5 n + 6)/(36 (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 2) (n + 3)

    (n + 4) (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/36*(39245*n^6+492989*n^5+2521869*n^4+6721947*n^3+9844846*n^2+7509424*n+
2328480)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+1)/(2*n+3
)/(5+2*n)/(7+2*n)/(n+2)/(n+3)/(n+4)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3
*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (4815125 n  + 82504266 n  + 610543234 n  + 2547429056 n  + 6551085513 n

                    3                2
     + 10626137878 n  + 10608760608 n  + 5954744160 n + 1437004800) (5 n + 1)

                                                                          /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (
                                                                        /

                           2                                    2
    576 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3) (n + 4) (n + 3)  (n + 2)

    (4 n + 9) (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/576*(4815125*n^8+82504266*n^7+610543234*n^6+2547429056*n^5+6551085513*n^4+
10626137878*n^3+10608760608*n^2+5954744160*n+1437004800)*(5*n+1)*(5*n+2)*(5*n+3
)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(7+2*n)/(5+2*n)^2/(2*n+1)/(2*n+3)/(n+4)
/(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (1967697 n  + 34328750 n  + 259154470 n  + 1105323708 n  + 2911880777 n

                   3               2
     + 4849092022 n  + 4981031616 n  + 2882582880 n + 718502400) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                           2                                    2
    576 (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3) (n + 4) (n + 3)  (n + 2)

    (4 n + 9) (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/576*(1967697*n^8+34328750*n^7+259154470*n^6+1105323708*n^5+2911880777*n^4+
4849092022*n^3+4981031616*n^2+2882582880*n+718502400)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(5+2*n)^2/(2*n+1)/(2*n+3)/(n+4)/
(n+3)^2/(n+2)/(4*n+9)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5             4              3
125 (125849 n  + 1980735 n  + 13189729 n  + 48123545 n  + 103768086 n

                  2
     + 132016120 n  + 91547136 n + 26611200) (5 n + 2) (5 n + 3) (5 n + 4)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (576
                                                                 /

                       2                                    2
    (7 + 2 n) (5 + 2 n)  (2 n + 1) (2 n + 3) (n + 4) (n + 3)  (n + 2) (4 n + 9)

    (3 n + 5) (3 n + 8) (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



125/576*(125849*n^7+1980735*n^6+13189729*n^5+48123545*n^4+103768086*n^3+
132016120*n^2+91547136*n+26611200)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*
n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(5+2*n)^2/(2*n+1)/(2*n+3)/(n+4)/(n+3)^2/(n
+2)/(4*n+9)/(3*n+5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   9              8               7
    (5 n + 2) (5 n + 1) (25485898 n  + 509921450 n  + 4490786079 n

                    6                5                 4                 3
     + 22838879103 n  + 73884827937 n  + 157589766375 n  + 221475683126 n

                     2                                   /
     + 197621750032 n  + 101497255200 n + 22832409600)  /  (576 (3 n + 10)
                                                       /

    (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9)

                   2                                                 2
    (n + 2) (n + 3)  (n + 4) (4 n + 11) (2 n + 3) (2 n + 1) (5 + 2 n)

    (7 + 2 n))



                             and in Maple notation



25/576*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
25485898*n^9+509921450*n^8+4490786079*n^7+22838879103*n^6+73884827937*n^5+
157589766375*n^4+221475683126*n^3+197621750032*n^2+101497255200*n+22832409600)/
(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+3)^
2/(n+4)/(4*n+11)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6              5               4
125 (1139381 n  + 20685925 n  + 162588233 n  + 722165049 n  + 1981197862 n

                   3               2
     + 3434680994 n  + 3670544044 n  + 2207477712 n + 570810240) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /             2
    (5 n + 6)  /  (288 (n + 3)  (n + 4) (2 n + 3) (4 n + 9) (4 n + 13)
              /

             2
    (5 + 2 n)  (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11)

    (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



125/288*(1139381*n^8+20685925*n^7+162588233*n^6+722165049*n^5+1981197862*n^4+
3434680994*n^3+3670544044*n^2+2207477712*n+570810240)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+3)^2/(n+4)/(2*n+3)/(4*n+9)/(
4*n+13)/(5+2*n)^2/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+
2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7               6                5
125 (4292346 n  + 89702638 n  + 827102797 n  + 4414118839 n  + 15017336863 n

                    4                3                2
     + 33749814547 n  + 50056950058 n  + 47189741016 n  + 25617446976 n

     + 6088642560) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9)

                                    /
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (576 (3 n + 10) (7 + 3 n) (4 + 3 n)
                                  /

                                                                       2
    (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 3)

                                                    2
    (n + 4) (4 n + 11) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



125/576*(4292346*n^9+89702638*n^8+827102797*n^7+4414118839*n^6+15017336863*n^5+
33749814547*n^4+50056950058*n^3+47189741016*n^2+25617446976*n+6088642560)*(5*n+
2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+10)/(7
+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+3)^2/(n+4)/(4*
n+11)/(2*n+3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7             6              5               4
125 (630593 n  + 11834167 n  + 96226542 n  + 442411764 n  + 1256573547 n

                   3               2
     + 2254771053 n  + 2491958278 n  + 1547535176 n + 412251840) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /
    (5 n + 7) (5 n + 6)  /  (288 (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11)
                        /

                                                            2
    (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 3)  (n + 4)

                                            2
    (4 n + 11) (2 n + 3) (2 n + 1) (5 + 2 n)  (7 + 2 n))



                             and in Maple notation



125/288*(630593*n^8+11834167*n^7+96226542*n^6+442411764*n^5+1256573547*n^4+
2254771053*n^3+2491958278*n^2+1547535176*n+412251840)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+12)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+10)/(7+3*n)/(4+3*
n)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+3)^2/(n+4)/(4*n+11)/(2*n+
3)/(2*n+1)/(5+2*n)^2/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4)

                                            8             7              6
    (5 n + 3) (5 n + 2) (5 n + 1) (2184866 n  + 39194281 n  + 304056550 n

                   5               4               3               2
     + 1331517695 n  + 3597775176 n  + 6137459520 n  + 6449284968 n

                                   /
     + 3812018784 n + 968647680)  /  (384 (4 n + 13) (4 n + 9) (3 n + 10)
                                 /

                                                              2
    (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 3)  (n + 4)

                                 2          2
    (2 n + 3) (2 n + 1) (7 + 2 n)  (5 + 2 n)  (4 n + 11))



                             and in Maple notation



125/384*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*
(5*n+1)*(2184866*n^8+39194281*n^7+304056550*n^6+1331517695*n^5+3597775176*n^4+
6137459520*n^3+6449284968*n^2+3812018784*n+968647680)/(4*n+13)/(4*n+9)/(3*n+10)
/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(n+3)^2/(n+4)/(2*n+3)/(2*n+1)/(7+2*n)
^2/(5+2*n)^2/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
125 (15669646 n  + 321527509 n  + 2909664257 n  + 15236513651 n

                    5                 4                 3                 2
     + 50858082881 n  + 112156692176 n  + 163295653648 n  + 151218966184 n

     + 80717857248 n + 18888629760) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11)

                                                        /
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1152 (4 n + 13)
                                                      /

    (4 n + 9) (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5)

           2                                      2          2
    (n + 3)  (n + 4) (2 n + 3) (2 n + 1) (7 + 2 n)  (5 + 2 n)  (4 n + 11))



                             and in Maple notation



125/1152*(15669646*n^9+321527509*n^8+2909664257*n^7+15236513651*n^6+50858082881
*n^5+112156692176*n^4+163295653648*n^3+151218966184*n^2+80717857248*n+
18888629760)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(
5*n+6)/(4*n+13)/(4*n+9)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(n+3)
^2/(n+4)/(2*n+3)/(2*n+1)/(7+2*n)^2/(5+2*n)^2/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7               6                5
125 (6567806 n  + 138591305 n  + 1290774347 n  + 6960492903 n  + 23934441309 n

                    4                3                2
     + 54380307300 n  + 81554110858 n  + 77743441612 n  + 42672130480 n

     + 10251521280) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 11)

                                                        /
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1152 (n + 2)
                                                      /

           2                                                 2
    (n + 3)  (n + 4) (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)  (2 n + 1)

                                                                           2
    (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n)

    (3 n + 11))



                             and in Maple notation



125/1152*(6567806*n^9+138591305*n^8+1290774347*n^7+6960492903*n^6+23934441309*n
^5+54380307300*n^4+81554110858*n^3+77743441612*n^2+42672130480*n+10251521280)*(
5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6
)/(n+2)/(n+3)^2/(n+4)/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)^2/(2*n+1)/(3*n+8)/(3*n+
10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6              5               4
125 (774170 n  + 14715241 n  + 121203264 n  + 564480193 n  + 1623988770 n

                   3               2
     + 2951131534 n  + 3301949116 n  + 2074835952 n + 558835200) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9)

                                    /                     2
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (384 (n + 2) (n + 3)  (n + 4) (2 n + 3)
                                  /

                                  2
    (4 n + 9) (4 n + 13) (5 + 2 n)  (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n)

                                            2
    (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/384*(774170*n^8+14715241*n^7+121203264*n^6+564480193*n^5+1623988770*n^4+
2951131534*n^3+3301949116*n^2+2074835952*n+558835200)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2
/(n+4)/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)^2/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n
+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 3, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
125 (16657294 n  + 363251185 n  + 3493112371 n  + 19427767479 n

                    5                 4                 3                 2
     + 68814676845 n  + 160824762780 n  + 247692622274 n  + 242053667996 n

     + 135930129776 n + 33337624320) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1152
                                                                 /

                   2                                                 2
    (n + 2) (n + 3)  (n + 4) (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)

    (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n)

             2
    (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/1152*(16657294*n^9+363251185*n^8+3493112371*n^7+19427767479*n^6+68814676845
*n^5+160824762780*n^4+247692622274*n^3+242053667996*n^2+135930129776*n+
33337624320)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*
(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)^2/(2*n+1)/
(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7             6             5              4              3
25 (751297 n  + 11201830 n  + 70240255 n  + 239780986 n  + 480598672 n

                  2
     + 564808576 n  + 359912064 n + 95800320) (5 n + 1) (5 n + 2) (5 n + 3)

                                            /
    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (216 (2 n + 1) (2 n + 3)
                                          /

                               2
    (5 + 2 n) (7 + 2 n) (n + 4)  (n + 2) (n + 3) (3 n + 5) (3 n + 8) (3 n + 11)

    (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/216*(751297*n^7+11201830*n^6+70240255*n^5+239780986*n^4+480598672*n^3+
564808576*n^2+359912064*n+95800320)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+4)^2/(n+2)/(n+3)/(3*n+5)/(3*n+8)/
(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5              4              3
25 (330025 n  + 5142781 n  + 33652801 n  + 119709451 n  + 249701974 n

                  2
     + 305124280 n  + 202066368 n + 55883520) (5 n + 1) (5 n + 2) (5 n + 3)

                                                      /
    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (144 (2 n + 1)
                                                    /

                                         2        2
    (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 3)  (n + 4)  (n + 2) (3 n + 5) (3 n + 8)

    (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/144*(330025*n^7+5142781*n^6+33652801*n^5+119709451*n^4+249701974*n^3+
305124280*n^2+202066368*n+55883520)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)*(5*n+8)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+3)^2/(n+4)^2/(n+2)/(3*n+5
)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                    7            6
25 (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (377675 n  + 6001703 n

                 5              4              3              2
     + 40157795 n  + 146504321 n  + 314422178 n  + 396596936 n  + 271932672 n

                                                           /
     + 78059520) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (432 (2 n + 1)
                                                         /

                                         2        2
    (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 3)  (n + 4)  (n + 2) (3 n + 5) (3 n + 8)

    (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/432*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(377675*n^7+6001703*n^6+40157795*n^5+
146504321*n^4+314422178*n^3+396596936*n^2+271932672*n+78059520)*(5*n+9)*(5*n+8)
*(5*n+7)*(5*n+6)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+3)^2/(n+4)^2/(n+2)/(3*n+5)/
(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10              9               8                7
125 (37419058 n   + 908391514 n  + 9849929007 n  + 62799957729 n

                     6                 5                  4                  3
     + 260600669673 n  + 735063920637 n  + 1426326055238 n  + 1878488905976 n

                      2
     + 1605360950304 n  + 802823750784 n + 178092794880) (5 n + 2) (5 n + 3)

                                                                             /
    (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /
                                                                           /

                         2        2                                         2
    (1728 (n + 2) (n + 3)  (n + 4)  (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)

    (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n)

    (7 + 2 n) (3 n + 11))



                             and in Maple notation



125/1728*(37419058*n^10+908391514*n^9+9849929007*n^8+62799957729*n^7+
260600669673*n^6+735063920637*n^5+1426326055238*n^4+1878488905976*n^3+
1605360950304*n^2+802823750784*n+178092794880)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)
*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)^2/(2*n+3)/(4*n+9)/
(4*n+13)/(5+2*n)^2/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7
+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    9                8                 7
    (5 n + 2) (5 n + 1) (799013563 n  + 16809453756 n  + 155384376768 n

                     6                  5                  4                  3
     + 827979128514 n  + 2801429448261 n  + 6238151127234 n  + 9136708930112 n

                      2                                      /
     + 8481528173376 n  + 4523784160896 n + 1054857323520)  /  (3456 (3 n + 10)
                                                           /

    (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9)

           2        2                                        2
    (n + 4)  (n + 3)  (5 + 2 n) (2 n + 3) (2 n + 1) (7 + 2 n)  (4 n + 11))



                             and in Maple notation



25/3456*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
799013563*n^9+16809453756*n^8+155384376768*n^7+827979128514*n^6+2801429448261*n
^5+6238151127234*n^4+9136708930112*n^3+8481528173376*n^2+4523784160896*n+
1054857323520)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+
9)/(n+4)^2/(n+3)^2/(5+2*n)/(2*n+3)/(2*n+1)/(7+2*n)^2/(4*n+11)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10              9               8                7
125 (19941354 n   + 487306849 n  + 5321125964 n  + 34177368532 n

                     6                 5                 4                  3
     + 142929819528 n  + 406431169995 n  + 795286999746 n  + 1056478413728 n

                     2
     + 910838575648 n  + 459545991296 n + 102838095360) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /                      2        2
    (5 n + 6)  /  (1152 (n + 2) (n + 3)  (n + 4)  (2 n + 3) (4 n + 9)
              /

                        2
    (4 n + 13) (5 + 2 n)  (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5)

                                  2
    (4 n + 11) (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/1152*(19941354*n^10+487306849*n^9+5321125964*n^8+34177368532*n^7+
142929819528*n^6+406431169995*n^5+795286999746*n^4+1056478413728*n^3+
910838575648*n^2+459545991296*n+102838095360)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*
(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)^2/(2*n+3)/
(4*n+9)/(4*n+13)/(5+2*n)^2/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4
+3*n)/(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
125 (18918382 n  + 421065133 n  + 4128031659 n  + 23381734257 n

                    5                 4                 3                 2
     + 84256488087 n  + 200123719602 n  + 312928779536 n  + 310169427728 n

     + 176490678336 n + 43812679680) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12)

    (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)

       /                      2        2
      /  (3456 (n + 2) (n + 3)  (n + 4)  (2 n + 3) (4 n + 9) (4 n + 13)
     /

             2
    (5 + 2 n)  (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11)

                       2
    (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/3456*(18918382*n^9+421065133*n^8+4128031659*n^7+23381734257*n^6+84256488087
*n^5+200123719602*n^4+312928779536*n^3+310169427728*n^2+176490678336*n+
43812679680)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)
*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)^2/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)
^2/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10               9                8                 7
125 (87937802 n   + 2164038977 n  + 23802755856 n  + 154041495378 n

                     6                  5                  4                  3
     + 649236695760 n  + 1860978531213 n  + 3671365213894 n  + 4917728870992 n

                      2
     + 4275273413088 n  + 2174926602240 n + 490657305600) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                      2        2
    (5 n + 7) (5 n + 6)  /  (3456 (n + 2) (n + 3)  (n + 4)  (2 n + 3) (4 n + 9)
                        /

                        2
    (4 n + 13) (5 + 2 n)  (2 n + 1) (4 n + 15) (3 n + 8) (3 n + 10) (7 + 3 n)

                                            2
    (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/3456*(87937802*n^10+2164038977*n^9+23802755856*n^8+154041495378*n^7+
649236695760*n^6+1860978531213*n^5+3671365213894*n^4+4917728870992*n^3+
4275273413088*n^2+2174926602240*n+490657305600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12
)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)
^2/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)^2/(2*n+1)/(4*n+15)/(3*n+8)/(3*n+10)/(7+3*n)
/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
125 (10907422 n  + 244930901 n  + 2422917443 n  + 13848424979 n

                    5                 4                 3                 2
     + 50356987523 n  + 120688483664 n  + 190401320812 n  + 190365520056 n

     + 109228380000 n + 27329702400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12)

    (5 n + 13) (5 n + 14) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /                      2        2
    (5 n + 6)  /  (1152 (n + 2) (n + 3)  (n + 4)  (2 n + 3) (4 n + 9)
              /

                        2
    (4 n + 13) (5 + 2 n)  (2 n + 1) (4 n + 15) (3 n + 8) (3 n + 10) (7 + 3 n)

                                            2
    (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/1152*(10907422*n^9+244930901*n^8+2422917443*n^7+13848424979*n^6+50356987523
*n^5+120688483664*n^4+190401320812*n^3+190365520056*n^2+109228380000*n+
27329702400)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+14)*(5*n+11)*(5*n+1
)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)^2/(2*n+3)/(4*n+9)/(4*n+13
)/(5+2*n)^2/(2*n+1)/(4*n+15)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/
(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 4, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



                10               9                8                 7
125 (239284858 n   + 6100111501 n  + 69458963118 n  + 464936319204 n

                      6                  5                   4
     + 2024740628166 n  + 5989662930639 n  + 12178507038242 n

                       3                   2
     + 16786947678896 n  + 14992281671616 n  + 7820148389760 n + 1805037696000)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1)

                                              /                      2        2
    (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (3456 (n + 2) (n + 3)  (n + 4)
                                            /

                                            2
    (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)  (2 n + 1) (4 n + 15) (3 n + 8)

                                                                 2
    (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n) (7 + 2 n)  (3 n + 11))



                             and in Maple notation



125/3456*(239284858*n^10+6100111501*n^9+69458963118*n^8+464936319204*n^7+
2024740628166*n^6+5989662930639*n^5+12178507038242*n^4+16786947678896*n^3+
14992281671616*n^2+7820148389760*n+1805037696000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+
12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+
4)^2/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)^2/(2*n+1)/(4*n+15)/(3*n+8)/(3*n+10)/(7+3*
n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4               3
25 (2607645 n  + 39552713 n  + 252428877 n  + 877574015 n  + 1792549110 n

                   2
     + 2148798872 n  + 1398249408 n + 380540160) (5 n + 1) (5 n + 2) (5 n + 3)

                                                      /
    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (432 (2 n + 1)
                                                    /

                                         2
    (2 n + 3) (5 + 2 n) (7 + 2 n) (n + 4)  (n + 2) (n + 3) (3 n + 5) (3 n + 8)

    (3 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10) (3 n + 13))



                             and in Maple notation



25/432*(2607645*n^7+39552713*n^6+252428877*n^5+877574015*n^4+1792549110*n^3+
2148798872*n^2+1398249408*n+380540160)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5
*n+6)*(5*n+7)*(5*n+8)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(n+4)^2/(n+2)/(n+3)/(3*n+
5)/(3*n+8)/(3*n+11)/(4+3*n)/(7+3*n)/(3*n+10)/(3*n+13)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                 7            6             5              4
    (5 n + 2) (5 n + 1) (318849 n  + 5055321 n  + 33667733 n  + 121933151 n

                  3              2                             /
     + 259069754 n  + 322646968 n  + 217918784 n + 61501440)  /  (72 (3 n + 13)
                                                             /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (3 n + 11) (3 n + 8) (3 n + 5) (n + 2)

           2        2
    (n + 4)  (n + 3)  (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1))



                             and in Maple notation



25/72*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
318849*n^7+5055321*n^6+33667733*n^5+121933151*n^4+259069754*n^3+322646968*n^2+
217918784*n+61501440)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(3*n+11)/(3*n+8)/(3*n+5
)/(n+2)/(n+4)^2/(n+3)^2/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5              4              3
25 (478008 n  + 7735663 n  + 52719821 n  + 195931269 n  + 428421019 n

                  2
     + 550604348 n  + 384631152 n + 112432320) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /            2        2
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (54 (n + 3)  (n + 4)
                                                      /

    (2 n + 3) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5)

    (4 + 3 n) (7 + 2 n) (3 n + 11) (3 n + 13))



                             and in Maple notation



25/54*(478008*n^7+7735663*n^6+52719821*n^5+195931269*n^4+428421019*n^3+
550604348*n^2+384631152*n+112432320)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5
*n+8)*(5*n+7)*(5*n+6)/(n+3)^2/(n+4)^2/(2*n+3)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/
(7+3*n)/(3*n+5)/(4+3*n)/(7+2*n)/(3*n+11)/(3*n+13)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10               9                8                 7
125 (63407711 n   + 1620579925 n  + 18477710748 n  + 123715876350 n

                     6                  5                  4                  3
     + 538380553275 n  + 1590173904885 n  + 3225954388762 n  + 4434343113080 n

                      2
     + 3947910387504 n  + 2052513677760 n + 472215744000) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /                      2        2
    (5 n + 6)  /  (1152 (n + 2) (n + 3)  (n + 4)  (2 n + 3) (4 n + 9)
              /

    (4 n + 13) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5)

                                  2
    (4 n + 11) (4 + 3 n) (7 + 2 n)  (3 n + 11) (3 n + 13))



                             and in Maple notation



125/1152*(63407711*n^10+1620579925*n^9+18477710748*n^8+123715876350*n^7+
538380553275*n^6+1590173904885*n^5+3225954388762*n^4+4434343113080*n^3+
3947910387504*n^2+2052513677760*n+472215744000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12
)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(n+4)^2/(2*n+3
)/(4*n+9)/(4*n+13)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4*n+11)/(4
+3*n)/(7+2*n)^2/(3*n+11)/(3*n+13)










Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 2, n + 5, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 14) (5 n + 11) (

               10               9                8                 7
    146089326 n   + 3831830471 n  + 44825197134 n  + 307817680854 n

                      6                  5                  4
     + 1373327978574 n  + 4156473926859 n  + 8634906218646 n

                       3                   2
     + 12145166696536 n  + 11053267909920 n  + 5867259223680 n + 1376171596800)

                                                        /
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1728 (n + 2)
                                                      /

           2        2                                         2
    (n + 3)  (n + 4)  (2 n + 3) (4 n + 9) (4 n + 13) (5 + 2 n)  (2 n + 1)

    (4 n + 15) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 n + 11) (4 + 3 n)

             2
    (7 + 2 n)  (3 n + 11) (3 n + 13))



                             and in Maple notation



125/1728*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+14)*(5*n+11)*(146089326
*n^10+3831830471*n^9+44825197134*n^8+307817680854*n^7+1373327978574*n^6+
4156473926859*n^5+8634906218646*n^4+12145166696536*n^3+11053267909920*n^2+
5867259223680*n+1376171596800)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n
+3)^2/(n+4)^2/(2*n+3)/(4*n+9)/(4*n+13)/(5+2*n)^2/(2*n+1)/(4*n+15)/(3*n+8)/(3*n+
10)/(7+3*n)/(3*n+5)/(4*n+11)/(4+3*n)/(7+2*n)^2/(3*n+11)/(3*n+13)














Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                    8             7
5 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (3110768 n  + 32900492 n

                  6              5              4              3              2
     + 143291780 n  + 335787329 n  + 462713204 n  + 383427221 n  + 186227622 n

                               /                                             2
     + 48291264 n + 5080320)  /  (128 (7 + 2 n) (4 n + 3) (5 + 4 n) (2 n + 1)
                             /

                                 2
    (5 + 2 n) (4 n + 7) (2 n + 3)  (4 n + 1) (2 n + 9) (n + 4))



                             and in Maple notation



5/128*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(3110768*n^8+32900492*n^7+143291780*n^6+
335787329*n^5+462713204*n^4+383427221*n^3+186227622*n^2+48291264*n+5080320)/(7+
2*n)/(4*n+3)/(5+4*n)/(2*n+1)^2/(5+2*n)/(4*n+7)/(2*n+3)^2/(4*n+1)/(2*n+9)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8              7               6               5
15 (11473008 n  + 153731324 n  + 878440396 n  + 2804056469 n  + 5497883470 n

                   4               3               2
     + 6849664151 n  + 5408792854 n  + 2602842456 n  + 689859072 n + 76204800)

                                                      /
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)  /  (128 (n + 4) (n + 2)
                                                    /

             2          2
    (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (5 + 2 n) (7 + 2 n)

    (2 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



15/128*(11473008*n^9+153731324*n^8+878440396*n^7+2804056469*n^6+5497883470*n^5+
6849664151*n^4+5408792854*n^3+2602842456*n^2+689859072*n+76204800)*(5*n+1)*(5*n
+2)*(5*n+3)*(5*n+4)*(1+n)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+
9)/(5+2*n)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9            8             7              6              5
125 (456784 n  + 6394260 n  + 38462364 n  + 130351239 n  + 273898212 n

                  4              3              2
     + 369300123 n  + 318622379 n  + 168938583 n  + 49640472 n + 6096384)

                                                        /
    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 6)  /  (128 (n + 4) (n + 2)
                                                      /

             2          2
    (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (5 + 2 n) (7 + 2 n)

    (2 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



125/128*(456784*n^9+6394260*n^8+38462364*n^7+130351239*n^6+273898212*n^5+
369300123*n^4+318622379*n^3+168938583*n^2+49640472*n+6096384)*(5*n+2)*(5*n+3)*(
5*n+4)*(5*n+1)*(5*n+6)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+9)/
(5+2*n)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7              6               5
25 (3300736 n  + 46580064 n  + 280881852 n  + 948010746 n  + 1969566456 n

                   4               3               2
     + 2606729061 n  + 2193138023 n  + 1128145254 n  + 320671008 n + 38102400)

                                                                /
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)  /  (64 (n + 4)
                                                              /

                     2          2          2
    (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9)

    (7 + 2 n) (2 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



25/64*(3300736*n^9+46580064*n^8+280881852*n^7+948010746*n^6+1969566456*n^5+
2606729061*n^4+2193138023*n^3+1128145254*n^2+320671008*n+38102400)*(5*n+1)*(5*n
+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*
n+1)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            9            8             7              6              5
75 (430336 n  + 6342240 n  + 40231644 n  + 143990838 n  + 319927026 n

                  4              3              2
     + 456671295 n  + 417567221 n  + 234821472 n  + 73193328 n + 9525600)

                                                                  /
    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (64 (n + 4)
                                                                /

                     2          2          2
    (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9)

    (7 + 2 n) (2 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



75/64*(430336*n^9+6342240*n^8+40231644*n^7+143990838*n^6+319927026*n^5+
456671295*n^4+417567221*n^3+234821472*n^2+73193328*n+9525600)*(5*n+2)*(5*n+3)*(
5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1
)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                      11               10
5 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (522997120 n   + 9409756864 n

                    9                 8                  7                  6
     + 75379215552 n  + 354812909088 n  + 1089899130402 n  + 2292363834165 n

                      5                  4                  3
     + 3364988431568 n  + 3441783053297 n  + 2398249334868 n

                      2                                   /
     + 1080563454756 n  + 281899336320 n + 32006016000)  /  (128 (n + 4)
                                                        /

                     2          2          2
    (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9)

    (7 + 2 n) (2 n + 9) (4 n + 3) (4 n + 7))



                             and in Maple notation



5/128*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(522997120*n^11+9409756864*n^10+
75379215552*n^9+354812909088*n^8+1089899130402*n^7+2292363834165*n^6+
3364988431568*n^5+3441783053297*n^4+2398249334868*n^3+1080563454756*n^2+
281899336320*n+32006016000)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(
5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7               6               5
75 (3870176 n  + 57026248 n  + 359502812 n  + 1270244818 n  + 2767194605 n

                   4               3               2
     + 3847639282 n  + 3408246633 n  + 1849827162 n  + 555768864 n + 69854400)

                                                                          /
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (
                                                                        /

                                 2          2          2
    128 (n + 4) (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (7 + 2 n) (2 n + 9) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/128*(3870176*n^9+57026248*n^8+359502812*n^7+1270244818*n^6+2767194605*n^5+
3847639282*n^4+3408246633*n^3+1849827162*n^2+555768864*n+69854400)*(5*n+1)*(5*n
+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^2/(5+2*
n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7               6               5
75 (3357248 n  + 51633872 n  + 342124624 n  + 1280112344 n  + 2975941264 n

                   4               3               2
     + 4448259443 n  + 4262148521 n  + 2512600236 n  + 820770408 n + 111767040)

                                                                            /
    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                                 2          2          2
    256 (n + 4) (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (7 + 2 n) (2 n + 9) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/256*(3357248*n^9+51633872*n^8+342124624*n^7+1280112344*n^6+2975941264*n^5+
4448259443*n^4+4262148521*n^3+2512600236*n^2+820770408*n+111767040)*(5*n+2)*(5*
n+3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^2/(5
+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        12                11
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (1995091168 n   + 40751033128 n

                     10                  9                  8
     + 375022763956 n   + 2055721187926 n  + 7472857293243 n

                       7                   6                   5
     + 18967560389895 n  + 34440364315232 n  + 45022194551159 n

                       4                   3                   2
     + 41985518040869 n  + 27177251889572 n  + 11553234218892 n

                                         /                               2
     + 2882369266560 n + 316859558400)  /  (128 (n + 4) (n + 2) (2 n + 1)
                                       /

             2          2
    (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9)

    (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



15/128*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(1995091168*n^12+40751033128*n^11+
375022763956*n^10+2055721187926*n^9+7472857293243*n^8+18967560389895*n^7+
34440364315232*n^6+45022194551159*n^5+41985518040869*n^4+27177251889572*n^3+
11553234218892*n^2+2882369266560*n+316859558400)/(n+4)/(n+2)/(2*n+1)^2/(2*n+3)^
2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        12                 11
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (6349606976 n   + 133167570448 n

                      10                  9                   8
     + 1259393689216 n   + 7099146736200 n  + 26551180299768 n

                       7                    6                    5
     + 69359080580619 n  + 129633305536318 n  + 174428879549615 n

                        4                    3                   2
     + 167395458252056 n  + 111463679882808 n  + 48714870135816 n

                                           /                               2
     + 12484851896160 n + 1408264704000)  /  (256 (n + 4) (n + 2) (2 n + 1)
                                         /

             2          2
    (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9)

    (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



15/256*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(6349606976*n^12+133167570448*n^11+
1259393689216*n^10+7099146736200*n^9+26551180299768*n^8+69359080580619*n^7+
129633305536318*n^6+174428879549615*n^5+167395458252056*n^4+111463679882808*n^3
+48714870135816*n^2+12484851896160*n+1408264704000)/(n+4)/(n+2)/(2*n+1)^2/(2*n+
3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8             7              6              5
525 (915440 n  + 13991292 n  + 91569336 n  + 336243765 n  + 762226911 n

                   4               3              2
     + 1104531195 n  + 1021322347 n  + 579494802 n  + 182174832 n + 23950080)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)

                /                                       2          2          2
    (5 n + 8)  /  (512 (n + 2) (n + 3) (n + 4) (2 n + 1)  (2 n + 3)  (5 + 2 n)
              /

    (4 n + 3) (4 n + 7) (4 n + 11) (7 + 2 n) (2 n + 9) (4 n + 1) (5 + 4 n)

    (4 n + 9))



                             and in Maple notation



525/512*(915440*n^9+13991292*n^8+91569336*n^7+336243765*n^6+762226911*n^5+
1104531195*n^4+1021322347*n^3+579494802*n^2+182174832*n+23950080)*(5*n+1)*(5*n+
2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(n+2)/(n+3)/(n+4)/(2*n+1)^2/(2
*n+3)^2/(5+2*n)^2/(4*n+3)/(4*n+7)/(4*n+11)/(7+2*n)/(2*n+9)/(4*n+1)/(5+4*n)/(4*n
+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7              6               5
125 (1795504 n  + 28625340 n  + 196724460 n  + 763886457 n  + 1844044884 n

                   4               3               2
     + 2863910727 n  + 2852507288 n  + 1748322780 n  + 593477280 n + 83825280)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /                                       2          2          2
    (5 n + 6)  /  (512 (n + 2) (n + 3) (n + 4) (2 n + 1)  (2 n + 3)  (5 + 2 n)
              /

    (4 n + 3) (4 n + 7) (4 n + 11) (7 + 2 n) (2 n + 9) (4 n + 1) (5 + 4 n)

    (4 n + 9))



                             and in Maple notation



125/512*(1795504*n^9+28625340*n^8+196724460*n^7+763886457*n^6+1844044884*n^5+
2863910727*n^4+2852507288*n^3+1748322780*n^2+593477280*n+83825280)*(5*n+2)*(5*n
+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)/(n+4)/(2*n+1)^2
/(2*n+3)^2/(5+2*n)^2/(4*n+3)/(4*n+7)/(4*n+11)/(7+2*n)/(2*n+9)/(4*n+1)/(5+4*n)/(
4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                        12                  11
5 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (59044578352 n   + 1238623452748 n

                       10                   9                    8
     + 11703478810584 n   + 65849255974533 n  + 245629161105615 n

                        7                     6                     5
     + 639580793773086 n  + 1191058799118668 n  + 1596499650840947 n

                         4                     3                    2
     + 1526205927756933 n  + 1012457103085566 n  + 440955561712248 n

                                             /
     + 112660820334720 n + 12674382336000)  /  (512 (7 + 2 n) (4 n + 11)
                                           /

                                 2          2                              2
    (4 n + 3) (5 + 4 n) (2 n + 1)  (5 + 2 n)  (4 n + 9) (4 n + 7) (2 n + 3)

    (4 n + 1) (2 n + 9) (n + 4) (n + 3))



                             and in Maple notation



5/512*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(59044578352*n^12+1238623452748*n^11+
11703478810584*n^10+65849255974533*n^9+245629161105615*n^8+639580793773086*n^7+
1191058799118668*n^6+1596499650840947*n^5+1526205927756933*n^4+1012457103085566
*n^3+440955561712248*n^2+112660820334720*n+12674382336000)/(7+2*n)/(4*n+11)/(4*
n+3)/(5+4*n)/(2*n+1)^2/(5+2*n)^2/(4*n+9)/(4*n+7)/(2*n+3)^2/(4*n+1)/(2*n+9)/(n+4
)/(n+3)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                         12                11
375 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (1464650256 n   + 31767717940 n

                     10                  9                  8
     + 310585120036 n   + 1809009728247 n  + 6986796936372 n

                       7                   6                   5
     + 18835043351190 n  + 36301789211520 n  + 50330226238115 n

                       4                   3                   2
     + 49725037146632 n  + 34054390464540 n  + 15291438987024 n

                                         /
     + 4021396823808 n + 464727352320)  /  (512 (7 + 2 n) (4 n + 11) (4 n + 3)
                                       /

                       2          2                              2
    (5 + 4 n) (2 n + 1)  (5 + 2 n)  (4 n + 9) (4 n + 7) (2 n + 3)  (4 n + 1)

    (2 n + 9) (n + 4) (n + 3))



                             and in Maple notation



375/512*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(1464650256*n^12+31767717940*n^11+
310585120036*n^10+1809009728247*n^9+6986796936372*n^8+18835043351190*n^7+
36301789211520*n^6+50330226238115*n^5+49725037146632*n^4+34054390464540*n^3+
15291438987024*n^2+4021396823808*n+464727352320)/(7+2*n)/(4*n+11)/(4*n+3)/(5+4*
n)/(2*n+1)^2/(5+2*n)^2/(4*n+9)/(4*n+7)/(2*n+3)^2/(4*n+1)/(2*n+9)/(n+4)/(n+3)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3         2
15 (223 n  + 1437 n  + 2768 n + 1512) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (1 + n)/(4 (n + 2) (n + 3) (n + 4) (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n)

    (2 n + 9))



                             and in Maple notation



15/4*(223*n^3+1437*n^2+2768*n+1512)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)/(n+2)
/(n+3)/(n+4)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5              4              3
25 (545164 n  + 7501992 n  + 43196029 n  + 134979408 n  + 247336825 n

                  2
     + 265912950 n  + 155397312 n + 38102400) (5 n + 1) (5 n + 2) (5 n + 3)

                                  /               2          2
    (5 n + 4) (1 + n) (5 n + 6)  /  (128 (2 n + 3)  (5 + 2 n)  (2 n + 1)
                                /

    (n + 2) (n + 3) (n + 4) (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9) (4 n + 7))



                             and in Maple notation



25/128*(545164*n^7+7501992*n^6+43196029*n^5+134979408*n^4+247336825*n^3+
265912950*n^2+155397312*n+38102400)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9
)/(4*n+7)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



           6           5            4            3            2
75 (29938 n  + 377037 n  + 1932388 n  + 5161329 n  + 7580284 n  + 5807304 n

     + 1814400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)

       /                                             2
      /  (128 (7 + 2 n) (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 9) (4 n + 7)
     /

    (2 n + 3) (2 n + 9) (n + 2) (n + 3) (n + 4))



                             and in Maple notation



75/128*(29938*n^6+377037*n^5+1932388*n^4+5161329*n^3+7580284*n^2+5807304*n+
1814400)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(7+2*n)/(5+4*n)/(2*n+1
)/(5+2*n)^2/(4*n+9)/(4*n+7)/(2*n+3)/(2*n+9)/(n+2)/(n+3)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5             4              3
75 (320588 n  + 4575048 n  + 27352613 n  + 88857588 n  + 169532801 n

                  2
     + 190171314 n  + 116278848 n + 29937600) (5 n + 1) (5 n + 2) (5 n + 3)

                                            /               2          2
    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (128 (2 n + 3)  (5 + 2 n)
                                          /

    (2 n + 1) (n + 2) (n + 3) (n + 4) (4 n + 7) (4 n + 11) (7 + 2 n) (2 n + 9)

    (5 + 4 n) (4 n + 9))



                             and in Maple notation



75/128*(320588*n^7+4575048*n^6+27352613*n^5+88857588*n^4+169532801*n^3+
190171314*n^2+116278848*n+29937600)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(4*n+7)/(4*n+11)/(7+2*
n)/(2*n+9)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5             4             3
75 (125548 n  + 1842336 n  + 11375449 n  + 38343546 n  + 76278640 n

                 2
     + 89657748 n  + 57719493 n + 15717240) (5 n + 2) (5 n + 3) (5 n + 4)

                                              /               2          2
    (5 n + 1) (5 n + 8) (5 n + 7) (5 n + 6)  /  (128 (2 n + 3)  (5 + 2 n)
                                            /

    (2 n + 1) (n + 2) (n + 3) (n + 4) (4 n + 7) (4 n + 11) (7 + 2 n) (2 n + 9)

    (5 + 4 n) (4 n + 9))



                             and in Maple notation



75/128*(125548*n^7+1842336*n^6+11375449*n^5+38343546*n^4+76278640*n^3+89657748*
n^2+57719493*n+15717240)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6
)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(4*n+7)/(4*n+11)/(7+2*n)/(2*n+9
)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            6            5            4             3             2
25 (119972 n  + 1612416 n  + 8866991 n  + 25548258 n  + 40695827 n

     + 33996936 n + 11642400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9)

                                    /               2          2
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (128 (2 n + 3)  (5 + 2 n)  (2 n + 1)
                                  /

    (n + 2) (n + 3) (n + 4) (4 n + 7) (4 n + 11) (7 + 2 n) (2 n + 9) (5 + 4 n)

    (4 n + 9))



                             and in Maple notation



25/128*(119972*n^6+1612416*n^5+8866991*n^4+25548258*n^3+40695827*n^2+33996936*n
+11642400)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n
+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(4*n+7)/(4*n+11)/(7+2*n)/(2*n+9)/(5+4
*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6            5             4             3
225 (87392 n  + 1286776 n  + 7945458 n  + 26688015 n  + 52723942 n

                 2
     + 61357277 n  + 39016980 n + 10478160) (5 n + 1) (5 n + 2) (5 n + 3)

                                                      /               2
    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (256 (2 n + 3)
                                                    /

             2                          2
    (5 + 2 n)  (2 n + 1) (n + 4) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (7 + 2 n)

    (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



225/256*(87392*n^7+1286776*n^6+7945458*n^5+26688015*n^4+52723942*n^3+61357277*n
^2+39016980*n+10478160)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(
5*n+8)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+2)/(5+4*n)/(4*n+9)/(7+2*n)/
(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5             4             3
75 (114240 n  + 1733808 n  + 11085130 n  + 38737781 n  + 80002568 n

                 2
     + 97772431 n  + 65558202 n + 18627840) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /               2
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (256 (2 n + 3)
                                                      /

             2                          2
    (5 + 2 n)  (2 n + 1) (n + 4) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (7 + 2 n)

    (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/256*(114240*n^7+1733808*n^6+11085130*n^5+38737781*n^4+80002568*n^3+97772431*
n^2+65558202*n+18627840)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7
)*(5*n+6)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+2)/(5+4*n)/(4*n+9)/(7+2*
n)/(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7             6             5              4              3
25 (645664 n  + 10308168 n  + 69647578 n  + 258280215 n  + 568025926 n

                  2
     + 741197577 n  + 531542232 n + 161663040) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /               2
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (256 (2 n + 3)
                                                      /

             2                          2
    (5 + 2 n)  (2 n + 1) (n + 4) (n + 3)  (n + 2) (5 + 4 n) (4 n + 9) (7 + 2 n)

    (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/256*(645664*n^7+10308168*n^6+69647578*n^5+258280215*n^4+568025926*n^3+
741197577*n^2+531542232*n+161663040)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5
*n+8)*(5*n+7)*(5*n+6)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(n+3)^2/(n+2)/(5+4*n)/(
4*n+9)/(7+2*n)/(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9              8               7               6                5
75 (8383456 n  + 166186248 n  + 1450768778 n  + 7322874369 n  + 23560361087 n

                    4                3                2
     + 50121322065 n  + 70521775231 n  + 63299733918 n  + 32897756448 n

     + 7544275200) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)

       /               2          2                          2
      /  (256 (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 4) (n + 3)  (n + 2)
     /

    (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/256*(8383456*n^9+166186248*n^8+1450768778*n^7+7322874369*n^6+23560361087*n^5
+50121322065*n^4+70521775231*n^3+63299733918*n^2+32897756448*n+7544275200)*(5*n
+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+4)/(
n+3)^2/(n+2)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                 6            5             4             3
    (5 n + 2) (5 n + 1) (609652 n  + 7394082 n  + 36379762 n  + 93344013 n

                  2                            /
     + 132294475 n  + 98578776 n + 30270240)  /  (128 (4 n + 11) (4 n + 7)
                                             /

    (2 n + 9) (7 + 2 n) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 4) (n + 3) (n + 2)

                       2          2
    (2 n + 1) (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



25/128*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
609652*n^6+7394082*n^5+36379762*n^4+93344013*n^3+132294475*n^2+98578776*n+
30270240)/(4*n+11)/(4*n+7)/(2*n+9)/(7+2*n)/(4*n+13)/(4*n+9)/(5+4*n)/(n+4)/(n+3)
/(n+2)/(2*n+1)/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             6            5             4             3             2
375 (190952 n  + 2409288 n  + 12409220 n  + 33518865 n  + 50241248 n

     + 39727467 n + 12972960) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9)

                                    /
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n) (4 n + 11) (5 + 4 n)
                                  /

                       2                                         2
    (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3)  (2 n + 9)

    (n + 3) (n + 4))



                             and in Maple notation



375/256*(190952*n^6+2409288*n^5+12409220*n^4+33518865*n^3+50241248*n^2+39727467
*n+12972960)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7
+2*n)/(4*n+11)/(5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*
n+9)/(n+3)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



            5          4           3           2
2625 (2858 n  + 32639 n  + 145991 n  + 320569 n  + 346298 n + 147420) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /                                                       2
    (5 n + 6)  /  (64 (7 + 2 n) (4 n + 11) (5 + 4 n) (2 n + 1) (5 + 2 n)
              /

                                            2
    (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3)  (2 n + 9) (n + 3) (n + 4))



                             and in Maple notation



2625/64*(2858*n^5+32639*n^4+145991*n^3+320569*n^2+346298*n+147420)*(5*n+2)*(5*n
+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(4*n+11)/(
5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*n+9)/(n+3)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9               8                7                 6
25 (275151224 n  + 5410684936 n  + 46901522932 n  + 235331763397 n

                     5                  4                  3                  2
     + 753527236897 n  + 1597261307899 n  + 2241966981743 n  + 2009858254908 n

     + 1044421323264 n + 239740300800) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1)

                          /
    (5 n + 7) (5 n + 6)  /  (256 (4 n + 11) (4 n + 7) (2 n + 9) (7 + 2 n)
                        /

                                                                              2
    (4 n + 13) (4 n + 9) (5 + 4 n) (n + 4) (n + 3) (n + 2) (2 n + 1) (5 + 2 n)

             2
    (2 n + 3) )



                             and in Maple notation



25/256*(275151224*n^9+5410684936*n^8+46901522932*n^7+235331763397*n^6+
753527236897*n^5+1597261307899*n^4+2241966981743*n^3+2009858254908*n^2+
1044421323264*n+239740300800)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(
4*n+11)/(4*n+7)/(2*n+9)/(7+2*n)/(4*n+13)/(4*n+9)/(5+4*n)/(n+4)/(n+3)/(n+2)/(2*n
+1)/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, 1 + n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8              7              6               5                4
75 (7029934 n  + 124046913 n  + 947363535 n  + 4092466233 n  + 10942865916 n

                    3                2
     + 18554570742 n  + 19490522495 n  + 11601068472 n + 2996753760) (5 n + 2)

                                                                   /
    (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 1) (5 n + 7) (5 n + 6)  /  (64
                                                                 /

    (4 n + 11) (4 n + 7) (2 n + 9) (7 + 2 n) (4 n + 13) (4 n + 9) (5 + 4 n)

                                               2          2
    (n + 4) (n + 3) (n + 2) (2 n + 1) (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



75/64*(7029934*n^8+124046913*n^7+947363535*n^6+4092466233*n^5+10942865916*n^4+
18554570742*n^3+19490522495*n^2+11601068472*n+2996753760)*(5*n+2)*(5*n+3)*(5*n+
4)*(5*n+12)*(5*n+1)*(5*n+7)*(5*n+6)/(4*n+11)/(4*n+7)/(2*n+9)/(7+2*n)/(4*n+13)/(
4*n+9)/(5+4*n)/(n+4)/(n+3)/(n+2)/(2*n+1)/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5             4              3
125 (114670 n  + 1980573 n  + 14428932 n  + 57445228 n  + 134895821 n

                  2
     + 186703208 n  + 140913360 n + 44706816) (5 n + 2) (5 n + 3) (5 n + 4)

                     2
    (5 n + 1) (1 + n)  (5 n + 6)/(18 (n + 2) (n + 3) (n + 4) (2 n + 9)

    (2 n + 3) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5)

    (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



125/18*(114670*n^7+1980573*n^6+14428932*n^5+57445228*n^4+134895821*n^3+
186703208*n^2+140913360*n+44706816)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(1+n)^2*(5*
n+6)/(n+2)/(n+3)/(n+4)/(2*n+9)/(2*n+3)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)
/(3*n+5)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5              4              3
25 (333226 n  + 5274343 n  + 34836508 n  + 124344721 n  + 258830818 n

                  2
     + 313988128 n  + 205426656 n + 55883520) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(24 (n + 2) (n + 3) (n + 4) (2 n + 9)

    (2 n + 3) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5)

    (4 + 3 n) (7 + 2 n) (3 n + 11))



                             and in Maple notation



25/24*(333226*n^7+5274343*n^6+34836508*n^5+124344721*n^4+258830818*n^3+
313988128*n^2+205426656*n+55883520)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)/(n+2)/(n+3)/(n+4)/(2*n+9)/(2*n+3)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(
7+3*n)/(3*n+5)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7           6            5            4             3             2
875 (19742 n  + 317483 n  + 2136773 n  + 7798127 n  + 16660037 n  + 20830118 n

     + 14108760 n + 3991680) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8)

    (5 n + 7) (5 n + 6)/(144 (n + 2) (n + 3) (n + 4) (2 n + 9) (2 n + 3)

    (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 + 3 n)

    (7 + 2 n) (3 n + 11))



                             and in Maple notation



875/144*(19742*n^7+317483*n^6+2136773*n^5+7798127*n^4+16660037*n^3+20830118*n^2
+14108760*n+3991680)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(n
+2)/(n+3)/(n+4)/(2*n+9)/(2*n+3)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5
)/(4+3*n)/(7+2*n)/(3*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10               9                8                 7
25 (203207495 n   + 4648916253 n  + 47235418686 n  + 280650158514 n

                      6                  5                  4
     + 1079654940471 n  + 2809442910573 n  + 5006948090644 n

                      3                  2
     + 6033085481796 n  + 4702287676464 n  + 2139898932864 n + 431532541440)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)

                /                     2
    (5 n + 8)  /  (576 (n + 4) (n + 3)  (n + 2) (4 n + 7) (4 n + 11) (3 n + 5)
              /

    (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 9)

    (4 + 3 n) (7 + 3 n) (3 n + 10) (4 n + 9))



                             and in Maple notation



25/576*(203207495*n^10+4648916253*n^9+47235418686*n^8+280650158514*n^7+
1079654940471*n^6+2809442910573*n^5+5006948090644*n^4+6033085481796*n^3+
4702287676464*n^2+2139898932864*n+431532541440)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)
*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(n+4)/(n+3)^2/(n+2)/(4*n+7)/(4*n+11)/(3*n+5)/(3*
n+8)/(3*n+11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)/
(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              10              9               8                7
25 (27267461 n   + 632089473 n  + 6516145500 n  + 39336630010 n

                     6                 5                 4                 3
     + 153983521553 n  + 408363624409 n  + 742921563270 n  + 915323697100 n

                     2
     + 730698523016 n  + 341144119008 n + 70692652800) (5 n + 2) (5 n + 3)

                                                                  /
    (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (192
                                                                /

                   2
    (n + 4) (n + 3)  (n + 2) (4 n + 7) (4 n + 11) (3 n + 5) (3 n + 8)

    (3 n + 11) (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n)

    (7 + 3 n) (3 n + 10) (4 n + 9))



                             and in Maple notation



25/192*(27267461*n^10+632089473*n^9+6516145500*n^8+39336630010*n^7+153983521553
*n^6+408363624409*n^5+742921563270*n^4+915323697100*n^3+730698523016*n^2+
341144119008*n+70692652800)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*
n+7)*(5*n+6)/(n+4)/(n+3)^2/(n+2)/(4*n+7)/(4*n+11)/(3*n+5)/(3*n+8)/(3*n+11)/(2*n
+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                             10               9
25 (5 n + 2) (5 n + 3) (5 n + 4) (128526659 n   + 3043844937 n

                    8                 7                 6                  5
     + 32096245930 n  + 198411780850 n  + 796135708747 n  + 2166056645641 n

                      4                  3                  2
     + 4045351172600 n  + 5118614045500 n  + 4196874686544 n  + 2012007900672 n

                                                                         /
     + 427844229120) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (576
                                                                       /

                   2
    (n + 4) (n + 3)  (n + 2) (4 n + 7) (4 n + 11) (3 n + 5) (3 n + 8)

    (3 n + 11) (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n)

    (7 + 3 n) (3 n + 10) (4 n + 9))



                             and in Maple notation



25/576*(5*n+2)*(5*n+3)*(5*n+4)*(128526659*n^10+3043844937*n^9+32096245930*n^8+
198411780850*n^7+796135708747*n^6+2166056645641*n^5+4045351172600*n^4+
5118614045500*n^3+4196874686544*n^2+2012007900672*n+427844229120)*(5*n+1)*(5*n+
9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+4)/(n+3)^2/(n+2)/(4*n+7)/(4*n+11)/(3*n+5)/(3*n+8)
/(3*n+11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)/(4*n
+9)










Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10               9                8                 7
125 (61642192 n   + 1513241454 n  + 16552445367 n  + 106203805928 n

                     6                  5                  4                  3
     + 442465086878 n  + 1250078414630 n  + 2424069130183 n  + 3183341772692 n

                      2
     + 2706929734100 n  + 1344353791536 n + 295679704320) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /
    (5 n + 6)  /  (1152 (3 n + 11) (7 + 2 n) (4 + 3 n) (4 n + 11) (3 n + 5)
              /

                                                      2
    (7 + 3 n) (3 n + 10) (3 n + 8) (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9)

                                                 2
    (4 n + 7) (2 n + 3) (2 n + 9) (n + 4) (n + 3)  (n + 2))



                             and in Maple notation



125/1152*(61642192*n^10+1513241454*n^9+16552445367*n^8+106203805928*n^7+
442465086878*n^6+1250078414630*n^5+2424069130183*n^4+3183341772692*n^3+
2706929734100*n^2+1344353791536*n+295679704320)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12
)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+11)/(7+2*n)/(4+3*n)/(4*
n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+
7)/(2*n+3)/(2*n+9)/(n+4)/(n+3)^2/(n+2)










Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 2, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10              9               8                7
125 (20932546 n   + 521297765 n  + 5785099870 n  + 37659491370 n

                     6                 5                 4                  3
     + 159179929998 n  + 456227826285 n  + 897327883130 n  + 1194915972580 n

                      2
     + 1029952442456 n  + 518226861600 n + 115399468800) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                          2
    (5 n + 7) (5 n + 6)  /  (384 (3 n + 11) (7 + 2 n)  (4 + 3 n) (4 n + 11)
                        /

                                                                2
    (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (2 n + 1) (5 + 2 n)  (4 n + 13)

                                                           2
    (4 n + 9) (4 n + 7) (2 n + 3) (2 n + 9) (n + 4) (n + 3)  (n + 2))



                             and in Maple notation



125/384*(20932546*n^10+521297765*n^9+5785099870*n^8+37659491370*n^7+
159179929998*n^6+456227826285*n^5+897327883130*n^4+1194915972580*n^3+
1029952442456*n^2+518226861600*n+115399468800)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)
*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+11)/(7+2*n)^2/(
4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*
n+9)/(4*n+7)/(2*n+3)/(2*n+9)/(n+4)/(n+3)^2/(n+2)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (2300197 n  + 42458602 n  + 336258769 n  + 1490876050 n  + 4043009602 n

                   3               2
     + 6858708364 n  + 7099287840 n  + 4094145216 n + 1005903360) (5 n + 1)

                                                                /             2
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (108 (n + 4)
                                                              /

    (n + 2) (n + 3) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/108*(2300197*n^8+42458602*n^7+336258769*n^6+1490876050*n^5+4043009602*n^4+
6858708364*n^3+7099287840*n^2+4094145216*n+1005903360)*(5*n+1)*(5*n+2)*(5*n+3)*
(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(n+4)^2/(n+2)/(n+3)/(3*n+5)/(3*n+8)/(3*n+11)/(2*n
+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            8            7             6              5              4
25 (450803 n  + 8572022 n  + 69951233 n  + 319602767 n  + 893188712 n

                   3               2
     + 1561718723 n  + 1666586172 n  + 991441728 n + 251475840) (5 n + 1)

                                                                          /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (36
                                                                        /

           2        2
    (n + 3)  (n + 4)  (n + 2) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1)

    (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/36*(450803*n^8+8572022*n^7+69951233*n^6+319602767*n^5+893188712*n^4+
1561718723*n^3+1666586172*n^2+991441728*n+251475840)*(5*n+1)*(5*n+2)*(5*n+3)*(5
*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(n+3)^2/(n+4)^2/(n+2)/(3*n+5)/(3*n+8)/(3*n+
11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (2006602 n  + 38685343 n  + 320715787 n  + 1492082983 n  + 4257000763 n

                   3               2
     + 7620373642 n  + 8350915968 n  + 5117526432 n + 1341204480) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

               2        2
    432 (n + 3)  (n + 4)  (n + 2) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1)

    (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



25/432*(2006602*n^8+38685343*n^7+320715787*n^6+1492082983*n^5+4257000763*n^4+
7620373642*n^3+8350915968*n^2+5117526432*n+1341204480)*(5*n+2)*(5*n+3)*(5*n+4)*
(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+3)^2/(n+4)^2/(n+2)/(3*n+5)/(3*n+8)/(
3*n+11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   10              9               8
    (5 n + 2) (5 n + 1) (29308856 n   + 705982080 n  + 7565263008 n

                    7                 6                 5                  4
     + 47472331227 n  + 193082534001 n  + 531581983737 n  + 1002669880237 n

                      3                  2
     + 1278573908718 n  + 1054097316648 n  + 506928220128 n + 107883135360)

       /
      /  (108 (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n)
     /

    (2 n + 3) (2 n + 1) (3 n + 11) (3 n + 8) (3 n + 5) (4 n + 13) (4 n + 9)

                   2        2
    (n + 2) (n + 4)  (n + 3)  (4 n + 11))



                             and in Maple notation



25/108*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
29308856*n^10+705982080*n^9+7565263008*n^8+47472331227*n^7+193082534001*n^6+
531581983737*n^5+1002669880237*n^4+1278573908718*n^3+1054097316648*n^2+
506928220128*n+107883135360)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(
2*n+3)/(2*n+1)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/(4*n+9)/(n+2)/(n+4)^2/(n+3)^2/
(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              10               9                8                 7
25 (81885984 n   + 1995065216 n  + 21649247000 n  + 137740263570 n

                     6                  5                  4                  3
     + 568793883307 n  + 1592229156173 n  + 3058337439525 n  + 3977790791105 n

                      2
     + 3350389134984 n  + 1648773369936 n + 359610451200) (5 n + 2) (5 n + 3)

                                                                  /
    (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (144
                                                                /

    (3 n + 11) (7 + 2 n) (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10)

    (3 n + 8) (2 n + 1) (5 + 2 n) (4 n + 13) (4 n + 9) (2 n + 3) (2 n + 9)

           2        2
    (n + 4)  (n + 3) )



                             and in Maple notation



25/144*(81885984*n^10+1995065216*n^9+21649247000*n^8+137740263570*n^7+
568793883307*n^6+1592229156173*n^5+3058337439525*n^4+3977790791105*n^3+
3350389134984*n^2+1648773369936*n+359610451200)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)
*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+11)/(7+2*n)/(4+3*n)/(4*n+11)/(3*n+5)/(7+3
*n)/(3*n+10)/(3*n+8)/(2*n+1)/(5+2*n)/(4*n+13)/(4*n+9)/(2*n+3)/(2*n+9)/(n+4)^2/(
n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8            7             6              5              4
125 (322160 n  + 6370544 n  + 54289856 n  + 260188754 n  + 766219751 n

                   3               2
     + 1418048921 n  + 1608364254 n  + 1020377736 n + 276623424) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9)

                                    /
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (216 (3 n + 10) (7 + 3 n) (4 + 3 n)
                                  /

    (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 11) (3 n + 8)

                                                  2        2
    (3 n + 5) (4 n + 13) (4 n + 9) (n + 2) (n + 4)  (n + 3)  (4 n + 11))



                             and in Maple notation



125/216*(322160*n^8+6370544*n^7+54289856*n^6+260188754*n^5+766219751*n^4+
1418048921*n^3+1608364254*n^2+1020377736*n+276623424)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+10)/(7+3
*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+11)/(3*n+8)/(3*n+5)/(4
*n+13)/(4*n+9)/(n+2)/(n+4)^2/(n+3)^2/(4*n+11)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7               6                5
625 (4477856 n  + 98472064 n  + 950572714 n  + 5284180717 n  + 18631707268 n

                    4                3                2
     + 43185298987 n  + 65752604186 n  + 63359213568 n  + 35023103136 n

     + 8449588224) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (3456
                                                                 /

                        2
    (3 n + 11) (7 + 2 n)  (4 + 3 n) (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10)

                                 2
    (3 n + 8) (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9) (2 n + 3) (2 n + 9)

           2        2
    (n + 4)  (n + 3) )



                             and in Maple notation



625/3456*(4477856*n^9+98472064*n^8+950572714*n^7+5284180717*n^6+18631707268*n^5
+43185298987*n^4+65752604186*n^3+63359213568*n^2+35023103136*n+8449588224)*(5*n
+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*
(5*n+6)/(3*n+11)/(7+2*n)^2/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(2
*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(2*n+3)/(2*n+9)/(n+4)^2/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10              9               8                7
125 (14335840 n   + 356187712 n  + 3946277830 n  + 25664506775 n

                     6                 5                 4                 3
     + 108451069490 n  + 310977595061 n  + 612388389760 n  + 817115184700 n

                     2
     + 706306861120 n  + 356704633152 n + 79801666560) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                           2
    (5 n + 7) (5 n + 6)  /  (1152 (3 n + 11) (7 + 2 n)  (4 + 3 n) (4 n + 11)
                        /

                                                                2
    (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (2 n + 1) (5 + 2 n)  (4 n + 13)

                                                 2        2
    (4 n + 9) (2 n + 3) (2 n + 9) (n + 2) (n + 4)  (n + 3) )



                             and in Maple notation



125/1152*(14335840*n^10+356187712*n^9+3946277830*n^8+25664506775*n^7+
108451069490*n^6+310977595061*n^5+612388389760*n^4+817115184700*n^3+
706306861120*n^2+356704633152*n+79801666560)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(
5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+11)/(7+2*n)^2/(4+
3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+
9)/(2*n+3)/(2*n+9)/(n+2)/(n+4)^2/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10               9                8                 7
125 (63625760 n   + 1599987392 n  + 17940082770 n  + 118060409925 n

                     6                  5                  4                  3
     + 504706073430 n  + 1463622002751 n  + 2913714327760 n  + 3928368998500 n

                      2
     + 3429133515360 n  + 1747748826432 n + 394314117120) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /                           2
    (5 n + 7) (5 n + 6)  /  (3456 (3 n + 11) (7 + 2 n)  (4 + 3 n) (4 n + 11)
                        /

                                                                2
    (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (2 n + 1) (5 + 2 n)  (4 n + 13)

                                                 2        2
    (4 n + 9) (2 n + 3) (2 n + 9) (n + 2) (n + 4)  (n + 3) )



                             and in Maple notation



125/3456*(63625760*n^10+1599987392*n^9+17940082770*n^8+118060409925*n^7+
504706073430*n^6+1463622002751*n^5+2913714327760*n^4+3928368998500*n^3+
3429133515360*n^2+1747748826432*n+394314117120)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12
)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+11)/(7+2*n)^2/
(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4
*n+9)/(2*n+3)/(2*n+9)/(n+2)/(n+4)^2/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



625 (5 n + 13) (5 n + 12) (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)

                                                              9             8
    (1 + n) (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (2082680 n  + 45401242 n

                  7               6               5                4
     + 434056411 n  + 2387462242 n  + 8321458942 n  + 19049002261 n

                    3                2                                 /
     + 28619943686 n  + 27193414440 n  + 14813438976 n + 3520661760)  /  (576
                                                                     /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (3 n + 11) (3 n + 8) (3 n + 5)

                                2        2                              2
    (4 n + 13) (4 n + 9) (n + 4)  (n + 3)  (2 n + 3) (2 n + 1) (7 + 2 n)

             2
    (5 + 2 n)  (4 n + 15) (4 n + 11))



                             and in Maple notation



625/576*(5*n+13)*(5*n+12)*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4
)*(5*n+3)*(5*n+2)*(5*n+1)*(2082680*n^9+45401242*n^8+434056411*n^7+2387462242*n^
6+8321458942*n^5+19049002261*n^4+28619943686*n^3+27193414440*n^2+14813438976*n+
3520661760)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(3*n+11)/(3*n+8)/(3*n+5)/(4*n+13)/
(4*n+9)/(n+4)^2/(n+3)^2/(2*n+3)/(2*n+1)/(7+2*n)^2/(5+2*n)^2/(4*n+15)/(4*n+11)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 3, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10              9                8                7
125 (38571376 n   + 979171780 n  + 11083926210 n  + 73637191260 n

                     6                 5                  4                  3
     + 317782909203 n  + 930190314930 n  + 1868817345875 n  + 2542185882710 n

                      2
     + 2238319796736 n  + 1150253053920 n + 261534873600) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 14) (5 n + 11) (5 n + 1) (5 n + 9)

                                    /                           2
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (1152 (3 n + 11) (7 + 2 n)  (4 + 3 n)
                                  /

    (4 n + 11) (3 n + 5) (7 + 3 n) (3 n + 10) (3 n + 8) (4 n + 15) (2 n + 1)

             2                                                 2        2
    (5 + 2 n)  (4 n + 13) (4 n + 9) (2 n + 3) (2 n + 9) (n + 4)  (n + 3)

    (n + 2))



                             and in Maple notation



125/1152*(38571376*n^10+979171780*n^9+11083926210*n^8+73637191260*n^7+
317782909203*n^6+930190314930*n^5+1868817345875*n^4+2542185882710*n^3+
2238319796736*n^2+1150253053920*n+261534873600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12
)*(5*n+13)*(5*n+14)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+11)/(
7+2*n)^2/(4+3*n)/(4*n+11)/(3*n+5)/(7+3*n)/(3*n+10)/(3*n+8)/(4*n+15)/(2*n+1)/(5+
2*n)^2/(4*n+13)/(4*n+9)/(2*n+3)/(2*n+9)/(n+4)^2/(n+3)^2/(n+2)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 4, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8              7               6               5                4
25 (8259294 n  + 154231013 n  + 1236396057 n  + 5552326001 n  + 15261263361 n

                    3                2
     + 26262297482 n  + 27601631208 n  + 16181964864 n + 4047563520) (5 n + 1)

                                                                          /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (
                                                                        /

               2
    216 (n + 4)  (n + 2) (n + 3) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1)

    (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10)

    (3 n + 13))



                             and in Maple notation



25/216*(8259294*n^8+154231013*n^7+1236396057*n^6+5552326001*n^5+15261263361*n^4
+26262297482*n^3+27601631208*n^2+16181964864*n+4047563520)*(5*n+1)*(5*n+2)*(5*n
+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(n+4)^2/(n+2)/(n+3)/(3*n+5)/(3*n+8)/(
3*n+11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)/(3*n+
13)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 4, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                 8             7              6              5
    (5 n + 2) (5 n + 1) (944421 n  + 18180105 n  + 150230033 n  + 695277251 n

                   4               3               2
     + 1969008434 n  + 3490428580 n  + 3778691912 n  + 2282091744 n + 588107520

        /
    )  /  (36 (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (7 + 2 n)
      /

    (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 11) (3 n + 8) (3 n + 5) (n + 2)

           2        2
    (n + 4)  (n + 3) )



                             and in Maple notation



25/36*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
944421*n^8+18180105*n^7+150230033*n^6+695277251*n^5+1969008434*n^4+3490428580*n
^3+3778691912*n^2+2282091744*n+588107520)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*
n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+11)/(3*n+8)/(3*n+5)/(n+2)/(n+4)^2/(n+
3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 4, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8              7               6                5
25 (22186854 n  + 433319437 n  + 3639757346 n  + 17159218488 n

                    4                3                 2
     + 49615811896 n  + 90023963723 n  + 100004580384 n  + 62122987152 n

     + 16501605120) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

                          /             2        2
    (5 n + 7) (5 n + 6)  /  (432 (n + 3)  (n + 4)  (2 n + 9) (2 n + 3)
                        /

    (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (7 + 3 n) (3 n + 5) (4 + 3 n)

    (7 + 2 n) (3 n + 11) (3 n + 13))



                             and in Maple notation



25/432*(22186854*n^8+433319437*n^7+3639757346*n^6+17159218488*n^5+49615811896*n
^4+90023963723*n^3+100004580384*n^2+62122987152*n+16501605120)*(5*n+2)*(5*n+3)*
(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+3)^2/(n+4)^2/(2*n+9)/(2*n+3)
/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(7+3*n)/(3*n+5)/(4+3*n)/(7+2*n)/(3*n+11)/(3*n
+13)




























Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   8              7               6
    (5 n + 2) (5 n + 1) (13757352 n  + 259902771 n  + 2108661218 n

                   5                4                3                2
     + 9587772879 n  + 26695446248 n  + 46562060124 n  + 49634898032 n

                                     /
     + 29538833856 n + 7507019520)  /  (216 (3 n + 13) (3 n + 10) (7 + 3 n)
                                   /

    (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14)

                                                          2
    (3 n + 11) (3 n + 8) (3 n + 5) (n + 3) (n + 2) (n + 4) )



                             and in Maple notation



25/216*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
13757352*n^8+259902771*n^7+2108661218*n^6+9587772879*n^5+26695446248*n^4+
46562060124*n^3+49634898032*n^2+29538833856*n+7507019520)/(3*n+13)/(3*n+10)/(7+
3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/
(3*n+5)/(n+3)/(n+2)/(n+4)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                  8             7              6
    (5 n + 2) (5 n + 1) (2892886 n  + 56362690 n  + 471455103 n

                   5               4                3                2
     + 2209076346 n  + 6335451289 n  + 11376981350 n  + 12481946272 n

                                    /
     + 7642961424 n + 1997835840)  /  (12 (3 n + 13) (3 n + 10) (7 + 3 n)
                                  /

    (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14)

                                          2        2
    (3 n + 11) (3 n + 8) (3 n + 5) (n + 4)  (n + 3) )



                             and in Maple notation



25/12*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
2892886*n^8+56362690*n^7+471455103*n^6+2209076346*n^5+6335451289*n^4+
11376981350*n^3+12481946272*n^2+7642961424*n+1997835840)/(3*n+13)/(3*n+10)/(7+3
*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(
3*n+5)/(n+4)^2/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 3, n + 5, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6               5               4
125 (2836282 n  + 56099551 n  + 477233903 n  + 2278634719 n  + 6673035863 n

                    3                2
     + 12262738354 n  + 13796082352 n  + 8678290656 n + 2333560320) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /
    (5 n + 6)  /  (144 (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9)
              /

    (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8)

                     2        2
    (3 n + 5) (n + 4)  (n + 3) )



                             and in Maple notation



125/144*(2836282*n^8+56099551*n^7+477233903*n^6+2278634719*n^5+6673035863*n^4+
12262738354*n^3+13796082352*n^2+8678290656*n+2333560320)*(5*n+2)*(5*n+3)*(5*n+4
)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+13)/(3*n+10)/(7+3*n)/(4
+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)
/(n+4)^2/(n+3)^2




























Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7               6               5
15 (22964912 n  + 300349452 n  + 1665492820 n  + 5129092961 n  + 9642228368 n

                    4               3               2
     + 11439733723 n  + 8535855280 n  + 3846422964 n  + 944183520 n + 95256000)

                                                      /               2
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n)  /  (128 (2 n + 1)
                                                    /

             2
    (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (5 + 2 n) (7 + 2 n) (2 n + 9)

    (4 n + 3) (4 n + 7) (n + 4) (n + 5))



                             and in Maple notation



15/128*(22964912*n^9+300349452*n^8+1665492820*n^7+5129092961*n^6+9642228368*n^5
+11439733723*n^4+8535855280*n^3+3846422964*n^2+944183520*n+95256000)*(5*n+1)*(5
*n+2)*(5*n+3)*(5*n+4)*(1+n)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+9)/(5+2*n)
/(7+2*n)/(2*n+9)/(4*n+3)/(4*n+7)/(n+4)/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7              6               5
175 (2268688 n  + 30890740 n  + 179032124 n  + 578740183 n  + 1147595732 n

                   4               3              2
     + 1444104265 n  + 1150035196 n  + 556962672 n  + 148089600 n + 16329600)

                                                                /
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6)  /  (256 (n + 4)
                                                              /

                             2          2
    (n + 5) (n + 2) (2 n + 1)  (2 n + 3)  (4 n + 3) (4 n + 7) (5 + 2 n)

    (7 + 2 n) (2 n + 9) (4 n + 1) (5 + 4 n) (4 n + 9))



                             and in Maple notation



175/256*(2268688*n^9+30890740*n^8+179032124*n^7+578740183*n^6+1147595732*n^5+
1444104265*n^4+1150035196*n^3+556962672*n^2+148089600*n+16329600)*(5*n+1)*(5*n+
2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(n+4)/(n+5)/(n+2)/(2*n+1)^2/(2*n+3)^2/(4*n+3)/
(4*n+7)/(5+2*n)/(7+2*n)/(2*n+9)/(4*n+1)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7              6               5
75 (1808592 n  + 25689572 n  + 156454700 n  + 535793731 n  + 1135512548 n

                   4               3              2
     + 1541274883 n  + 1335849020 n  + 709676474 n  + 208208880 n + 25401600)

                                                                  /
    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (256
                                                                /

                                     2          2
    (n + 4) (n + 5) (n + 2) (2 n + 1)  (2 n + 3)  (4 n + 3) (4 n + 7) (5 + 2 n)

    (7 + 2 n) (2 n + 9) (4 n + 1) (5 + 4 n) (4 n + 9))



                             and in Maple notation



75/256*(1808592*n^9+25689572*n^8+156454700*n^7+535793731*n^6+1135512548*n^5+
1541274883*n^4+1335849020*n^3+709676474*n^2+208208880*n+25401600)*(5*n+2)*(5*n+
3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(n+4)/(n+5)/(n+2)/(2*n+1)^2/(2*n+3)^2/(4*n+3
)/(4*n+7)/(5+2*n)/(7+2*n)/(2*n+9)/(4*n+1)/(5+4*n)/(4*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7               6               5
75 (2644856 n  + 43034836 n  + 301269590 n  + 1187406193 n  + 2893266884 n

                   4               3               2
     + 4500126199 n  + 4443593130 n  + 2666504952 n  + 872739360 n + 116424000)

                                                                          /
    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (
                                                                        /

                                         2          2          2
    128 (n + 4) (n + 5) (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1)

    (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/128*(2644856*n^9+43034836*n^8+301269590*n^7+1187406193*n^6+2893266884*n^5+
4500126199*n^4+4443593130*n^3+2666504952*n^2+872739360*n+116424000)*(5*n+1)*(5*
n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(n+4)/(n+5)/(n+2)/(2*n+1)^2/(2*n+3)^
2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              10             9              8              7               6
525 (1214656 n   + 21417168 n  + 165258352 n  + 733685968 n  + 2071660820 n

                   5               4               3               2
     + 3878978137 n  + 4864375048 n  + 4019841457 n  + 2084598424 n

     + 608015970 n + 74844000) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1)

                                    /
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (256 (n + 2) (n + 4) (n + 5) (2 n + 9)
                                  /

                       2                              2          2
    (4 n + 1) (2 n + 3)  (4 n + 7) (4 n + 9) (5 + 2 n)  (2 n + 1)  (5 + 4 n)

    (4 n + 3) (4 n + 11) (7 + 2 n))



                             and in Maple notation



525/256*(1214656*n^10+21417168*n^9+165258352*n^8+733685968*n^7+2071660820*n^6+
3878978137*n^5+4864375048*n^4+4019841457*n^3+2084598424*n^2+608015970*n+
74844000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+4)/(
n+5)/(2*n+9)/(4*n+1)/(2*n+3)^2/(4*n+7)/(4*n+9)/(5+2*n)^2/(2*n+1)^2/(5+4*n)/(4*n
+3)/(4*n+11)/(7+2*n)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7               6               5
125 (5816032 n  + 86167696 n  + 545330864 n  + 1932114724 n  + 4216714856 n

                   4               3               2
     + 5868343651 n  + 5196427624 n  + 2814196593 n  + 841311360 n + 104781600)

    (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)

                /                                                           2
    (5 n + 8)  /  (256 (n + 2) (n + 4) (n + 5) (2 n + 9) (4 n + 1) (2 n + 3)
              /

                                 2          2
    (4 n + 7) (4 n + 9) (5 + 2 n)  (2 n + 1)  (5 + 4 n) (4 n + 3) (4 n + 11)

    (7 + 2 n))



                             and in Maple notation



125/256*(5816032*n^9+86167696*n^8+545330864*n^7+1932114724*n^6+4216714856*n^5+
5868343651*n^4+5196427624*n^3+2814196593*n^2+841311360*n+104781600)*(5*n+1)*(5*
n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(n+2)/(n+4)/(n+5)/(2*n+9)/(4
*n+1)/(2*n+3)^2/(4*n+7)/(4*n+9)/(5+2*n)^2/(2*n+1)^2/(5+4*n)/(4*n+3)/(4*n+11)/(7
+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7               6                5
25 (12931936 n  + 199710416 n  + 1326281280 n  + 4966061628 n  + 11536795884 n

                    4                3               2
     + 17207313549 n  + 16424049530 n  + 9624674887 n  + 3116669490 n

     + 419126400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8)

                          /
    (5 n + 7) (5 n + 6)  /  (256 (n + 2) (n + 4) (n + 5) (2 n + 9) (4 n + 1)
                        /

             2                              2          2
    (2 n + 3)  (4 n + 7) (4 n + 9) (5 + 2 n)  (2 n + 1)  (5 + 4 n) (4 n + 3)

    (4 n + 11) (7 + 2 n))



                             and in Maple notation



25/256*(12931936*n^9+199710416*n^8+1326281280*n^7+4966061628*n^6+11536795884*n^
5+17207313549*n^4+16424049530*n^3+9624674887*n^2+3116669490*n+419126400)*(5*n+2
)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+4)/(n+5)/(2*
n+9)/(4*n+1)/(2*n+3)^2/(4*n+7)/(4*n+9)/(5+2*n)^2/(2*n+1)^2/(5+4*n)/(4*n+3)/(4*n
+11)/(7+2*n)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                         12                 11
15 (5 n + 1) (5 n + 4) (5 n + 3) (5 n + 2) (26330774048 n   + 538585989488 n

                      10                   9                   8
     + 4957255883712 n   + 27147910220916 n  + 98495220752412 n

                        7                    6                    5
     + 249282218365743 n  + 450924843569158 n  + 586703440434634 n

                        4                    3                    2
     + 544028964613740 n  + 349774415249619 n  + 147502710611730 n

                                           /
     + 36449453854800 n + 3960744480000)  /  (256 (n + 4) (n + 5) (2 n + 9)
                                         /

                       2                              2          2
    (4 n + 1) (2 n + 3)  (4 n + 7) (4 n + 9) (5 + 2 n)  (2 n + 1)  (5 + 4 n)

    (4 n + 3) (4 n + 11) (7 + 2 n))



                             and in Maple notation



15/256*(5*n+1)*(5*n+4)*(5*n+3)*(5*n+2)*(26330774048*n^12+538585989488*n^11+
4957255883712*n^10+27147910220916*n^9+98495220752412*n^8+249282218365743*n^7+
450924843569158*n^6+586703440434634*n^5+544028964613740*n^4+349774415249619*n^3
+147502710611730*n^2+36449453854800*n+3960744480000)/(n+4)/(n+5)/(2*n+9)/(4*n+1
)/(2*n+3)^2/(4*n+7)/(4*n+9)/(5+2*n)^2/(2*n+1)^2/(5+4*n)/(4*n+3)/(4*n+11)/(7+2*n
)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    10               9                8
    (5 n + 2) (5 n + 1) (100297552 n   + 1857221060 n  + 14977548852 n

                    7                 6                 5                 4
     + 69130618551 n  + 201819543072 n  + 388572851667 n  + 498544404416 n

                     3                 2                                 /
     + 419739790006 n  + 221103805944 n  + 65417734080 n + 8172964800)  /  (512
                                                                       /

    (4 n + 13) (4 n + 9) (5 + 4 n) (4 n + 1) (2 n + 9) (7 + 2 n) (4 n + 11)

                                 2          2          2
    (4 n + 7) (4 n + 3) (5 + 2 n)  (2 n + 3)  (2 n + 1)  (n + 5) (n + 4)

    (n + 3) (n + 2))



                             and in Maple notation



25/512*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
100297552*n^10+1857221060*n^9+14977548852*n^8+69130618551*n^7+201819543072*n^6+
388572851667*n^5+498544404416*n^4+419739790006*n^3+221103805944*n^2+65417734080
*n+8172964800)/(4*n+13)/(4*n+9)/(5+4*n)/(4*n+1)/(2*n+9)/(7+2*n)/(4*n+11)/(4*n+7
)/(4*n+3)/(5+2*n)^2/(2*n+3)^2/(2*n+1)^2/(n+5)/(n+4)/(n+3)/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10              9               8                7
125 (47932624 n   + 918247492 n  + 7699593228 n  + 37154541027 n

                     6                 5                 4                 3
     + 114061950450 n  + 232286543748 n  + 316957994852 n  + 285095144543 n

                     2
     + 160908557406 n  + 51030450630 n + 6810804000) (5 n + 2) (5 n + 3)

                                                                  /
    (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (512
                                                                /

                                     2          2          2
    (n + 3) (n + 4) (n + 5) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 3)

    (4 n + 7) (4 n + 11) (7 + 2 n) (2 n + 9) (4 n + 1) (5 + 4 n) (4 n + 9)

    (4 n + 13))



                             and in Maple notation



125/512*(47932624*n^10+918247492*n^9+7699593228*n^8+37154541027*n^7+
114061950450*n^6+232286543748*n^5+316957994852*n^4+285095144543*n^3+
160908557406*n^2+51030450630*n+6810804000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n
+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+3)/(n+4)/(n+5)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*
n+3)/(4*n+7)/(4*n+11)/(7+2*n)/(2*n+9)/(4*n+1)/(5+4*n)/(4*n+9)/(4*n+13)










Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



         3        2
175 (47 n  + 314 n  + 629 n + 360) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (1 + n) (5 n + 6)/(8 (n + 2) (n + 3) (n + 4) (n + 5) (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (2 n + 9))



                             and in Maple notation



175/8*(47*n^3+314*n^2+629*n+360)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/
(n+2)/(n+3)/(n+4)/(n+5)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5              4              3
75 (471766 n  + 7177529 n  + 45804909 n  + 158794497 n  + 322633085 n

                  2
     + 383772694 n  + 247248720 n + 66528000) (5 n + 1) (5 n + 2) (5 n + 3)

                                            /               2
    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (128 (5 + 2 n)  (2 n + 1)
                                          /

    (2 n + 3) (n + 2) (n + 3) (n + 4) (n + 5) (5 + 4 n) (4 n + 9) (7 + 2 n)

    (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/128*(471766*n^7+7177529*n^6+45804909*n^5+158794497*n^4+322633085*n^3+
383772694*n^2+247248720*n+66528000)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)/(5+2*n)^2/(2*n+1)/(2*n+3)/(n+2)/(n+3)/(n+4)/(n+5)/(5+4*n)/(4*n+9)/(7
+2*n)/(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7           6            5            4             3             2
525 (22986 n  + 357291 n  + 2337877 n  + 8345291 n  + 17545307 n  + 21719458 n

     + 14655390 n + 4158000) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 8)

                          /               2
    (5 n + 7) (5 n + 6)  /  (128 (5 + 2 n)  (2 n + 1) (2 n + 3) (n + 2) (n + 3)
                        /

    (n + 4) (n + 5) (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



525/128*(22986*n^7+357291*n^6+2337877*n^5+8345291*n^4+17545307*n^3+21719458*n^2
+14655390*n+4158000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+8)*(5*n+7)*(5*n+6)/(5
+2*n)^2/(2*n+1)/(2*n+3)/(n+2)/(n+3)/(n+4)/(n+5)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)
/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7             6             5              4              3
75 (877596 n  + 12649012 n  + 76310161 n  + 250044633 n  + 481159163 n

                  2
     + 544468235 n  + 335944800 n + 87318000) (5 n + 1) (5 n + 2) (5 n + 3)

                                                      /               2
    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (256 (2 n + 3)
                                                    /

             2
    (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (5 + 4 n) (4 n + 9)

    (7 + 2 n) (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/256*(877596*n^7+12649012*n^6+76310161*n^5+250044633*n^4+481159163*n^3+
544468235*n^2+335944800*n+87318000)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)*(5*n+8)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(n+5)/(5+4*n)/
(4*n+9)/(7+2*n)/(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



            7            6             5              4              3
75 (354460 n  + 5249236 n  + 32671417 n  + 110922521 n  + 222132933 n

                  2
     + 262716123 n  + 170110510 n + 46569600) (5 n + 2) (5 n + 3) (5 n + 4)

                                                        /               2
    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (256 (2 n + 3)
                                                      /

             2
    (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (n + 4) (n + 5) (5 + 4 n) (4 n + 9)

    (7 + 2 n) (2 n + 9) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/256*(354460*n^7+5249236*n^6+32671417*n^5+110922521*n^4+222132933*n^3+
262716123*n^2+170110510*n+46569600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*
n+8)*(5*n+7)*(5*n+6)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n+4)/(n+5)/(5+4*n
)/(4*n+9)/(7+2*n)/(2*n+9)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             6            5             4             3              2
125 (350804 n  + 4752468 n  + 26296499 n  + 76126437 n  + 121683857 n

     + 101885535 n + 34927200) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1)

                                              /
    (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n) (4 n + 11)
                                            /

                                 2                              2
    (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 9) (4 n + 7) (2 n + 3)  (2 n + 9)

    (n + 3) (n + 4) (n + 5))



                             and in Maple notation



125/256*(350804*n^6+4752468*n^5+26296499*n^4+76126437*n^3+121683857*n^2+
101885535*n+34927200)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(
5*n+6)/(7+2*n)/(4*n+11)/(5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*
n+9)/(n+3)/(n+4)/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                  8             7              6
    (5 n + 2) (5 n + 1) (4534664 n  + 81767704 n  + 633149744 n

                   5               4                3                2
     + 2749445911 n  + 7324484780 n  + 12263121157 n  + 12611143860 n

                                    /
     + 7290669780 n + 1816214400)  /  (256 (4 n + 11) (4 n + 7) (2 n + 9)
                                  /

                                                            2
    (7 + 2 n) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (n + 5) (n + 4)

                       2          2
    (2 n + 1) (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



25/256*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
4534664*n^8+81767704*n^7+633149744*n^6+2749445911*n^5+7324484780*n^4+
12263121157*n^3+12611143860*n^2+7290669780*n+1816214400)/(4*n+11)/(4*n+7)/(2*n+
9)/(7+2*n)/(4*n+13)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(n+5)/(n+4)/(2*n+1)/(5+2*n)^2
/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (2539766 n  + 46918757 n  + 373433567 n  + 1672772015 n  + 4613941529 n

                   3               2
     + 8028490703 n  + 8611899438 n  + 5210237025 n + 1362160800) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                                                         2
    64 (7 + 2 n) (4 n + 11) (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9)

                       2                                  2
    (4 n + 7) (2 n + 3)  (2 n + 9) (n + 5) (n + 4) (n + 3) )



                             and in Maple notation



25/64*(2539766*n^8+46918757*n^7+373433567*n^6+1672772015*n^5+4613941529*n^4+
8028490703*n^3+8611899438*n^2+5210237025*n+1362160800)*(5*n+2)*(5*n+3)*(5*n+4)*
(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(4*n+11)/(5+4*n)/(2*n+1)/(5+2*n
)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*n+9)/(n+5)/(n+4)/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6              5               4
125 (790312 n  + 15199904 n  + 126400504 n  + 593646545 n  + 1722390778 n

                   3               2
     + 3161572331 n  + 3585788136 n  + 2297677590 n + 636854400) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /
    (5 n + 6)  /  (256 (4 n + 11) (4 n + 7) (2 n + 9) (7 + 2 n) (4 n + 13)
              /

                                       2                                    2
    (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (n + 5) (n + 4) (2 n + 1) (5 + 2 n)

             2
    (2 n + 3) )



                             and in Maple notation



125/256*(790312*n^8+15199904*n^7+126400504*n^6+593646545*n^5+1722390778*n^4+
3161572331*n^3+3585788136*n^2+2297677590*n+636854400)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(4*n+11)/(4*n+7)/(2*n+9)/(7+2*n
)/(4*n+13)/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(n+5)/(n+4)/(2*n+1)/(5+2*n)^2/(2*n+3)^
2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 4, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



               11               10                9                 8
25 (199141334 n   + 5066159771 n   + 58138401570 n  + 397317427299 n

                      7                  6                   5
     + 1796831665095 n  + 5646855606141 n  + 12584959083835 n

                       4                   3                   2
     + 19892192480086 n  + 21855604969326 n  + 15897854062503 n

     + 6891058139040 n + 1348539192000) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1)

                          /
    (5 n + 7) (5 n + 6)  /  (64 (4 n + 11) (4 n + 7) (2 n + 9) (7 + 2 n)
                        /

                                                  2
    (4 n + 13) (4 n + 9) (5 + 4 n) (n + 2) (n + 3)  (n + 5) (n + 4) (2 n + 1)

             2          2
    (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



25/64*(199141334*n^11+5066159771*n^10+58138401570*n^9+397317427299*n^8+
1796831665095*n^7+5646855606141*n^6+12584959083835*n^5+19892192480086*n^4+
21855604969326*n^3+15897854062503*n^2+6891058139040*n+1348539192000)*(5*n+2)*(5
*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(4*n+11)/(4*n+7)/(2*n+9)/(7+2*n)/(4*n+13)
/(4*n+9)/(5+4*n)/(n+2)/(n+3)^2/(n+5)/(n+4)/(2*n+1)/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



175 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                  8             7              6
    (5 n + 2) (5 n + 1) (2640016 n  + 49051052 n  + 391391212 n

                   5               4               3               2
     + 1751782385 n  + 4811978284 n  + 8312971553 n  + 8829742848 n

                                    /
     + 5279103450 n + 1362160800)  /  (256 (4 n + 11) (4 n + 7) (2 n + 9)
                                  /

                                          2
    (4 n + 13) (4 n + 9) (5 + 4 n) (n + 3)  (n + 5) (n + 4) (2 n + 1)

             2          2          2
    (7 + 2 n)  (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



175/256*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
2640016*n^8+49051052*n^7+391391212*n^6+1751782385*n^5+4811978284*n^4+8312971553
*n^3+8829742848*n^2+5279103450*n+1362160800)/(4*n+11)/(4*n+7)/(2*n+9)/(4*n+13)/
(4*n+9)/(5+4*n)/(n+3)^2/(n+5)/(n+4)/(2*n+1)/(7+2*n)^2/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6               5               4
125 (1781072 n  + 33939964 n  + 278696024 n  + 1288309105 n  + 3668489633 n

                   3               2
     + 6593396776 n  + 7310148021 n  + 4574917305 n + 1238328000) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /
    (5 n + 6)  /  (256 (4 n + 11) (4 n + 7) (2 n + 9) (4 n + 13) (4 n + 9)
              /

                     2                                    2          2
    (5 + 4 n) (n + 3)  (n + 5) (n + 4) (2 n + 1) (7 + 2 n)  (5 + 2 n)

             2
    (2 n + 3) )



                             and in Maple notation



125/256*(1781072*n^8+33939964*n^7+278696024*n^6+1288309105*n^5+3668489633*n^4+
6593396776*n^3+7310148021*n^2+4574917305*n+1238328000)*(5*n+2)*(5*n+3)*(5*n+4)*
(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(4*n+11)/(4*n+7)/(2*n+9)/(4*n+
13)/(4*n+9)/(5+4*n)/(n+3)^2/(n+5)/(n+4)/(2*n+1)/(7+2*n)^2/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                           8             7              6
125 (5 n + 2) (5 n + 3) (5 n + 4) (768368 n  + 15235796 n  + 130637596 n

                  5               4               3               2
     + 632702195 n  + 1893365002 n  + 3585437234 n  + 4196531774 n

     + 2775959235 n + 794593800) (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9)

                                    /               2
    (5 n + 8) (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n)  (4 n + 11) (5 + 4 n)
                                  /

                       2                                         2
    (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3)  (2 n + 9)

                           2
    (n + 5) (n + 4) (n + 3)  (n + 2))



                             and in Maple notation



125/256*(5*n+2)*(5*n+3)*(5*n+4)*(768368*n^8+15235796*n^7+130637596*n^6+
632702195*n^5+1893365002*n^4+3585437234*n^3+4196531774*n^2+2775959235*n+
794593800)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)^2/
(4*n+11)/(5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*n+9)/(
n+5)/(n+4)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 5, n + 3],

      divided by the number of SYT of shape [n,n,n,n,n] is



                12                11                 10                  9
75 (1514861040 n   + 43358715348 n   + 565052082768 n   + 4433745155491 n

                       8                   7                    6
     + 23331124624533 n  + 86743994963838 n  + 233663742819846 n

                        5                    4                    3
     + 459503446956999 n  + 654738460622337 n  + 659254641370244 n

                        2
     + 445270295178276 n  + 181135441529280 n + 33563642112000) (5 n + 2)

                                                        /               2
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n)
                                                      /

                                            2
    (4 n + 11) (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9) (4 n + 7)

             2                                  2
    (2 n + 3)  (2 n + 9) (n + 5) (n + 4) (n + 3)  (n + 2))



                             and in Maple notation



75/256*(1514861040*n^12+43358715348*n^11+565052082768*n^10+4433745155491*n^9+
23331124624533*n^8+86743994963838*n^7+233663742819846*n^6+459503446956999*n^5+
654738460622337*n^4+659254641370244*n^3+445270295178276*n^2+181135441529280*n+
33563642112000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*n+6)/(7+2*n)^2/(4*n+
11)/(5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*n+9)/(n+5)/
(n+4)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, 1 + n, n + 5, n + 4],

      divided by the number of SYT of shape [n,n,n,n,n] is



                12                 11                  10                   9
25 (7202336720 n   + 209820010620 n   + 2783407113832 n   + 22233019385253 n

                        8                    7                     6
     + 119096539620609 n  + 450723897935307 n  + 1235717514884842 n

                         5                     4                     3
     + 2472868990375737 n  + 3584867271778151 n  + 3671492139264363 n

                         2
     + 2521586472057846 n  + 1042746622510320 n + 196347306355200) (5 n + 2)

                                                        /               2
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 7) (5 n + 6)  /  (256 (7 + 2 n)
                                                      /

                                            2
    (4 n + 11) (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9) (4 n + 7)

             2                                  2
    (2 n + 3)  (2 n + 9) (n + 5) (n + 4) (n + 3)  (n + 2))



                             and in Maple notation



25/256*(7202336720*n^12+209820010620*n^11+2783407113832*n^10+22233019385253*n^9
+119096539620609*n^8+450723897935307*n^7+1235717514884842*n^6+2472868990375737*
n^5+3584867271778151*n^4+3671492139264363*n^3+2521586472057846*n^2+
1042746622510320*n+196347306355200)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+7)*(5*
n+6)/(7+2*n)^2/(4*n+11)/(5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n
+3)^2/(2*n+9)/(n+5)/(n+4)/(n+3)^2/(n+2)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 2, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6               5               4
125 (1946446 n  + 35926997 n  + 284339047 n  + 1259117885 n  + 3408536299 n

                   3               2
     + 5769466238 n  + 5955775248 n  + 3423915360 n + 838252800) (5 n + 1)

    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(216 (n + 2)

    (n + 3) (n + 4) (n + 5) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



125/216*(1946446*n^8+35926997*n^7+284339047*n^6+1259117885*n^5+3408536299*n^4+
5769466238*n^3+5955775248*n^2+3423915360*n+838252800)*(5*n+1)*(5*n+2)*(5*n+3)*(
5*n+4)*(1+n)*(5*n+6)*(5*n+7)/(n+2)/(n+3)/(n+4)/(n+5)/(3*n+5)/(3*n+8)/(3*n+11)/(
2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 2, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5             4              3
175 (218896 n  + 3500411 n  + 23365441 n  + 84321839 n  + 177562855 n

                  2
     + 218076518 n  + 144595800 n + 39916800) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)/(72 (n + 2) (n + 3) (n + 4)

    (n + 5) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3) (5 + 2 n)

    (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



175/72*(218896*n^7+3500411*n^6+23365441*n^5+84321839*n^4+177562855*n^3+
218076518*n^2+144595800*n+39916800)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+
6)*(5*n+7)*(5*n+8)/(n+2)/(n+3)/(n+4)/(n+5)/(3*n+5)/(3*n+8)/(3*n+11)/(2*n+1)/(2*
n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 2, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7            6             5              4              3
125 (329677 n  + 5355791 n  + 36418897 n  + 134305949 n  + 290003818 n

                  2
     + 366560396 n  + 251065968 n + 71850240) (5 n + 2) (5 n + 3) (5 n + 4)

    (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)/(216 (n + 2) (n + 3)

    (n + 4) (n + 5) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3)

    (5 + 2 n) (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10))



                             and in Maple notation



125/216*(329677*n^7+5355791*n^6+36418897*n^5+134305949*n^4+290003818*n^3+
366560396*n^2+251065968*n+71850240)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*
n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)/(n+4)/(n+5)/(3*n+5)/(3*n+8)/(3*n+11)/(2*n+1)/(
2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 2, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



                11                10                 9                  8
125 (550513219 n   + 14875368227 n   + 181024436700 n  + 1309342865580 n

                      7                   6                   5
     + 6252632655927 n  + 20693158116951 n  + 48413739966290 n

                       4                   3                   2
     + 80040604974370 n  + 91593420134904 n  + 69052810028472 n

     + 30844984433760 n + 6178761388800) (5 n + 2) (5 n + 3) (5 n + 4)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (1728
                                                                 /

                   2
    (n + 2) (n + 3)  (n + 4) (n + 5) (2 n + 9) (2 n + 3) (4 n + 7) (4 n + 9)

    (4 n + 13) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (4 n + 11) (4 + 3 n)

    (7 + 2 n) (7 + 3 n) (3 n + 5) (3 n + 11))



                             and in Maple notation



125/1728*(550513219*n^11+14875368227*n^10+181024436700*n^9+1309342865580*n^8+
6252632655927*n^7+20693158116951*n^6+48413739966290*n^5+80040604974370*n^4+
91593420134904*n^3+69052810028472*n^2+30844984433760*n+6178761388800)*(5*n+2)*(
5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+2)/(n+3)^2/(
n+4)/(n+5)/(2*n+9)/(2*n+3)/(4*n+7)/(4*n+9)/(4*n+13)/(5+2*n)/(2*n+1)/(3*n+8)/(3*
n+10)/(4*n+11)/(4+3*n)/(7+2*n)/(7+3*n)/(3*n+5)/(3*n+11)






















Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7                6
25 (26637027 n  + 610126162 n  + 6089504720 n  + 34733378776 n

                     5                 4                 3                 2
     + 124666584083 n  + 291734534878 n  + 444669241850 n  + 425275016184 n

     + 231345184320 n + 54486432000) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                                            /                     2
    (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (216 (n + 5) (n + 4)  (n + 2)
                                          /

    (n + 3) (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3) (5 + 2 n)

    (7 + 2 n) (2 n + 9) (4 + 3 n) (7 + 3 n) (3 n + 10) (3 n + 13))



                             and in Maple notation



25/216*(26637027*n^9+610126162*n^8+6089504720*n^7+34733378776*n^6+124666584083*
n^5+291734534878*n^4+444669241850*n^3+425275016184*n^2+231345184320*n+
54486432000)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(n+5
)/(n+4)^2/(n+2)/(n+3)/(3*n+5)/(3*n+8)/(3*n+11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/
(2*n+9)/(4+3*n)/(7+3*n)/(3*n+10)/(3*n+13)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



175 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                 9             8              7               6
    (5 n + 2) (5 n + 1) (806217 n  + 18900351 n  + 193221428 n  + 1129583394 n

                   5               4                3                2
     + 4157604333 n  + 9981543359 n  + 15615534742 n  + 15336131776 n

                                    /
     + 8572024800 n + 2075673600)  /  (72 (3 n + 13) (3 n + 10) (7 + 3 n)
                                  /

    (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 11)

                                       2        2
    (3 n + 8) (3 n + 5) (n + 2) (n + 4)  (n + 3)  (n + 5))



                             and in Maple notation



175/72*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
806217*n^9+18900351*n^8+193221428*n^7+1129583394*n^6+4157604333*n^5+9981543359*
n^4+15615534742*n^3+15336131776*n^2+8572024800*n+2075673600)/(3*n+13)/(3*n+10)/
(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+11)/(3*n+8)/(3*n+5
)/(n+2)/(n+4)^2/(n+3)^2/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             9             8              7               6                5
25 (4067919 n  + 96444917 n  + 998691476 n  + 5924168378 n  + 22168955831 n

                    4                3                2
     + 54230568473 n  + 86651768294 n  + 87134344632 n  + 49990682880 n

     + 12454041600) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 6) (5 n + 7)

                          /            2        2
    (5 n + 8) (5 n + 9)  /  (27 (n + 3)  (n + 4)  (n + 5) (2 n + 9) (2 n + 3)
                        /

    (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (4 + 3 n) (7 + 2 n) (7 + 3 n)

    (3 n + 5) (3 n + 13) (3 n + 11))



                             and in Maple notation



25/27*(4067919*n^9+96444917*n^8+998691476*n^7+5924168378*n^6+22168955831*n^5+
54230568473*n^4+86651768294*n^3+87134344632*n^2+49990682880*n+12454041600)*(5*n
+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(n+3)^2/(n+4)^2/(n+
5)/(2*n+9)/(2*n+3)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(4+3*n)/(7+2*n)/(7+3*n)/(3*
n+5)/(3*n+13)/(3*n+11)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 3, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



625 (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (

             10             9              8               7                6
    1669976 n   + 44357108 n  + 523443830 n  + 3612656717 n  + 16142282633 n

                    5                 4                 3                 2
     + 48768228947 n  + 100824296085 n  + 140740723308 n  + 126826203396 n

     + 66541579920 n + 15411876480) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)

       /                     2        2
      /  (144 (n + 2) (n + 3)  (n + 4)  (n + 5) (2 n + 9) (2 n + 3) (4 n + 9)
     /

    (4 n + 13) (5 + 2 n) (2 n + 1) (3 n + 8) (3 n + 10) (4 n + 11) (4 + 3 n)

    (7 + 2 n) (7 + 3 n) (3 n + 5) (3 n + 13) (3 n + 11))



                             and in Maple notation



625/144*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(1669976*n^
10+44357108*n^9+523443830*n^8+3612656717*n^7+16142282633*n^6+48768228947*n^5+
100824296085*n^4+140740723308*n^3+126826203396*n^2+66541579920*n+15411876480)*(
5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(n+2)/(n+3)^2/(n+4)^2/(n+5)/(2*n+9)/(2*n+3)/(4*n
+9)/(4*n+13)/(5+2*n)/(2*n+1)/(3*n+8)/(3*n+10)/(4*n+11)/(4+3*n)/(7+2*n)/(7+3*n)/
(3*n+5)/(3*n+13)/(3*n+11)






















Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 4, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   9              8               7
    (5 n + 2) (5 n + 1) (22700157 n  + 523743660 n  + 5268633626 n

                    6                 5                 4                 3
     + 30307595152 n  + 109781372933 n  + 259446383620 n  + 399683883044 n

                     2                                   /
     + 386679272128 n  + 212998524480 n + 50854003200)  /  (108 (3 n + 13)
                                                       /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3)

    (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 3) (n + 2)

           2
    (n + 4)  (n + 5))



                             and in Maple notation



25/108*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
22700157*n^9+523743660*n^8+5268633626*n^7+30307595152*n^6+109781372933*n^5+
259446383620*n^4+399683883044*n^3+386679272128*n^2+212998524480*n+50854003200)/
(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+
14)/(3*n+11)/(3*n+8)/(3*n+5)/(n+3)/(n+2)/(n+4)^2/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 4, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



175 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                  9             8              7
    (5 n + 2) (5 n + 1) (2601013 n  + 61404075 n  + 632418344 n

                   6                5                4                3
     + 3726361418 n  + 13830499457 n  + 33500483435 n  + 52908360226 n

                    2                                 /
     + 52490289152 n  + 29658206880 n + 7264857600)  /  (24 (3 n + 13)
                                                    /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3)

                                                               2        2
    (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 4)  (n + 3)

    (n + 5))



                             and in Maple notation



175/24*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
2601013*n^9+61404075*n^8+632418344*n^7+3726361418*n^6+13830499457*n^5+
33500483435*n^4+52908360226*n^3+52490289152*n^2+29658206880*n+7264857600)/(3*n+
13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(
3*n+11)/(3*n+8)/(3*n+5)/(n+4)^2/(n+3)^2/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 4, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7                6
125 (8796464 n  + 210063045 n  + 2191735802 n  + 13104453854 n

                    5                 4                 3                 2
     + 49444141916 n  + 121991370365 n  + 196655182938 n  + 199558809056 n

     + 115559072160 n + 29059430400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (144 (3 n + 13)
                                                      /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3)

                                                               2        2
    (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 4)  (n + 3)

    (n + 5))



                             and in Maple notation



125/144*(8796464*n^9+210063045*n^8+2191735802*n^7+13104453854*n^6+49444141916*n
^5+121991370365*n^4+196655182938*n^3+199558809056*n^2+115559072160*n+
29059430400)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(
5*n+9)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1
)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(n+4)^2/(n+3)^2/(n+5)




























Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   9              8               7
    (5 n + 2) (5 n + 1) (24252415 n  + 563951229 n  + 5719879896 n

                    6                 5                 4                 3
     + 33188649786 n  + 121314926823 n  + 289466639073 n  + 450480839522 n

                     2                                   /             2
     + 440548567176 n  + 245475572160 n + 59329670400)  /  (216 (n + 5)
                                                       /

           2
    (n + 4)  (n + 3) (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9)

    (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8)

    (3 n + 5))



                             and in Maple notation



125/216*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
24252415*n^9+563951229*n^8+5719879896*n^7+33188649786*n^6+121314926823*n^5+
289466639073*n^4+450480839522*n^3+440548567176*n^2+245475572160*n+59329670400)/
(n+5)^2/(n+4)^2/(n+3)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)
/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4)

                                            9             8              7
    (5 n + 3) (5 n + 2) (5 n + 1) (3177715 n  + 75612177 n  + 785076796 n

                   6                5                4                3
     + 4664573264 n  + 17462539369 n  + 42678218315 n  + 68034516840 n

                    2                                 /
     + 68157913604 n  + 38905018320 n + 9631440000)  /  (36 (3 n + 13)
                                                    /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3)

                                                               2        2
    (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 5)  (n + 4)

           2
    (n + 3) )



                             and in Maple notation



125/36*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(
5*n+1)*(3177715*n^9+75612177*n^8+785076796*n^7+4664573264*n^6+17462539369*n^5+
42678218315*n^4+68034516840*n^3+68157913604*n^2+38905018320*n+9631440000)/(3*n+
13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(
3*n+11)/(3*n+8)/(3*n+5)/(n+5)^2/(n+4)^2/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 4, n + 5, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7                6
125 (7969282 n  + 191874729 n  + 2018684573 n  + 12172196920 n

                    5                 4                 3                 2
     + 46322582936 n  + 115289777605 n  + 187499390209 n  + 191969282266 n

     + 112159341000 n + 28454025600) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (216
                                                                 /

    (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 9) (7 + 2 n) (5 + 2 n)

                                                                         2
    (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 5)

           2        2
    (n + 4)  (n + 3) )



                             and in Maple notation



125/216*(7969282*n^9+191874729*n^8+2018684573*n^7+12172196920*n^6+46322582936*n
^5+115289777605*n^4+187499390209*n^3+191969282266*n^2+112159341000*n+
28454025600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*
(5*n+8)*(5*n+9)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+
3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(n+5)^2/(n+4)^2/(n+3)^2




























Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9              8               7                6
25 (54733072 n  + 727805556 n  + 4098033360 n  + 12801143331 n

                    5                4                3                2
     + 24386072937 n  + 29290083705 n  + 22100113583 n  + 10053876168 n

     + 2485204848 n + 251475840) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

                        /               2          2
    (1 + n) (5 n + 6)  /  (256 (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n)
                      /

    (4 n + 9) (5 + 2 n) (7 + 2 n) (2 n + 9) (2 n + 11) (4 n + 3) (4 n + 7)

    (n + 4) (n + 5))



                             and in Maple notation



25/256*(54733072*n^9+727805556*n^8+4098033360*n^7+12801143331*n^6+24386072937*n
^5+29290083705*n^4+22100113583*n^3+10053876168*n^2+2485204848*n+251475840)*(5*n
+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(
4*n+9)/(5+2*n)/(7+2*n)/(2*n+9)/(2*n+11)/(4*n+3)/(4*n+7)/(n+4)/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              10               9                8                7
75 (86945664 n   + 1439543824 n  + 10344779596 n  + 42389866188 n

                     6                 5                 4                 3
     + 109400530461 n  + 185268667254 n  + 207823996179 n  + 151911448682 n

                    2
     + 68931705320 n  + 17422428672 n + 1844156160) (5 n + 1) (5 n + 2)

                                                      /
    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)  /  (512 (n + 4) (n + 5)
                                                    /

                     2          2
    (n + 2) (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (5 + 2 n)

    (7 + 2 n) (2 n + 9) (2 n + 11) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/512*(86945664*n^10+1439543824*n^9+10344779596*n^8+42389866188*n^7+
109400530461*n^6+185268667254*n^5+207823996179*n^4+151911448682*n^3+68931705320
*n^2+17422428672*n+1844156160)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5
*n+7)/(n+4)/(n+5)/(n+2)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+9)/(5+2*n)/(7+
2*n)/(2*n+9)/(2*n+11)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



                                                      10              9
75 (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 1) (30439936 n   + 521615824 n

                   8                7                6                5
     + 3900277732 n  + 16731779672 n  + 45519695743 n  + 81878420003 n

                    4                3                2
     + 98339340483 n  + 77582916553 n  + 38277971586 n  + 10582325028 n

                                                   /
     + 1229437440) (5 n + 8) (5 n + 7) (5 n + 6)  /  (512 (n + 4) (n + 5)
                                                 /

                     2          2
    (n + 2) (2 n + 1)  (2 n + 3)  (4 n + 1) (5 + 4 n) (4 n + 9) (5 + 2 n)

    (7 + 2 n) (2 n + 9) (2 n + 11) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/512*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(30439936*n^10+521615824*n^9+3900277732*
n^8+16731779672*n^7+45519695743*n^6+81878420003*n^5+98339340483*n^4+77582916553
*n^3+38277971586*n^2+10582325028*n+1229437440)*(5*n+8)*(5*n+7)*(5*n+6)/(n+4)/(n
+5)/(n+2)/(2*n+1)^2/(2*n+3)^2/(4*n+1)/(5+4*n)/(4*n+9)/(5+2*n)/(7+2*n)/(2*n+9)/(
2*n+11)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              10               9                8                7
75 (89884640 n   + 1542891848 n  + 11516506788 n  + 49119354730 n

                     6                 5                 4                 3
     + 132256599989 n  + 234294814717 n  + 275757872817 n  + 212202399425 n

                     2
     + 101737397246 n  + 27272500200 n + 3073593600) (5 n + 1) (5 n + 2)

                                                                /
    (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (512 (n + 4)
                                                              /

                             2          2          2
    (n + 5) (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1) (5 + 4 n)

    (4 n + 9) (7 + 2 n) (2 n + 9) (2 n + 11) (4 n + 3) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/512*(89884640*n^10+1542891848*n^9+11516506788*n^8+49119354730*n^7+
132256599989*n^6+234294814717*n^5+275757872817*n^4+212202399425*n^3+
101737397246*n^2+27272500200*n+3073593600)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n
)*(5*n+6)*(5*n+7)*(5*n+8)/(n+4)/(n+5)/(n+2)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+
1)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(2*n+11)/(4*n+3)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              10              9               8                7
75 (36954976 n   + 656470152 n  + 5097055916 n  + 22744059918 n

                    6                 5                 4                 3
     + 64474369467 n  + 121056912468 n  + 152035278149 n  + 125634924522 n

                    2
     + 65024068432 n  + 18880078800 n + 2305195200) (5 n + 2) (5 n + 3)

                                                                  /
    (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (512
                                                                /

                                     2          2          2
    (n + 4) (n + 5) (n + 2) (2 n + 1)  (2 n + 3)  (5 + 2 n)  (4 n + 1)

    (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9) (2 n + 11) (4 n + 3) (4 n + 7)

    (4 n + 11))



                             and in Maple notation



75/512*(36954976*n^10+656470152*n^9+5097055916*n^8+22744059918*n^7+64474369467*
n^6+121056912468*n^5+152035278149*n^4+125634924522*n^3+65024068432*n^2+
18880078800*n+2305195200)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+
7)*(5*n+6)/(n+4)/(n+5)/(n+2)/(2*n+1)^2/(2*n+3)^2/(5+2*n)^2/(4*n+1)/(5+4*n)/(4*n
+9)/(7+2*n)/(2*n+9)/(2*n+11)/(4*n+3)/(4*n+7)/(4*n+11)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    10               9                8
    (5 n + 2) (5 n + 1) (511457504 n   + 9280165992 n  + 73155944548 n

                     7                 6                  5                  4
     + 329241727602 n  + 934814961885 n  + 1745704784313 n  + 2165985639547 n

                      3                 2                                   /
     + 1757808110703 n  + 889343791706 n  + 251745258600 n + 29967537600)  /  (
                                                                          /

    512 (4 n + 13) (4 n + 11) (4 n + 7) (4 n + 3) (2 n + 11) (2 n + 9)

                                                     2          2          2
    (7 + 2 n) (4 n + 9) (5 + 4 n) (4 n + 1) (5 + 2 n)  (2 n + 3)  (2 n + 1)

    (n + 2) (n + 5) (n + 4))



                             and in Maple notation



25/512*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
511457504*n^10+9280165992*n^9+73155944548*n^8+329241727602*n^7+934814961885*n^6
+1745704784313*n^5+2165985639547*n^4+1757808110703*n^3+889343791706*n^2+
251745258600*n+29967537600)/(4*n+13)/(4*n+11)/(4*n+7)/(4*n+3)/(2*n+11)/(2*n+9)/
(7+2*n)/(4*n+9)/(5+4*n)/(4*n+1)/(5+2*n)^2/(2*n+3)^2/(2*n+1)^2/(n+2)/(n+5)/(n+4)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10               9               8                7
525 (55325152 n   + 1037747176 n  + 8499820404 n  + 39970126626 n

                     6                 5                 4                 3
     + 119294092845 n  + 235624441509 n  + 311068529366 n  + 270026850779 n

                     2
     + 146698781373 n  + 44663648370 n + 5708102400) (5 n + 2) (5 n + 3)

                                                                  /
    (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (512
                                                                /

                                                                 2          2
    (7 + 2 n) (2 n + 11) (4 n + 11) (4 n + 3) (5 + 4 n) (2 n + 1)  (5 + 2 n)

                                            2
    (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3)  (4 n + 1) (2 n + 9) (n + 4)

    (n + 5))



                             and in Maple notation



525/512*(55325152*n^10+1037747176*n^9+8499820404*n^8+39970126626*n^7+
119294092845*n^6+235624441509*n^5+311068529366*n^4+270026850779*n^3+
146698781373*n^2+44663648370*n+5708102400)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+1)*(5*n
+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+11)/(4*n+11)/(4*n+3)/(5+4*n)/(2*n+1)^2
/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(4*n+1)/(2*n+9)/(n+4)/(n+5)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                     11                10                 9
    (5 n + 2) (5 n + 1) (2282702336 n   + 50249775456 n   + 488426050876 n

                      8                   7                   6
     + 2761367267964 n  + 10067671833429 n  + 24797768386797 n

                       5                   4                   3
     + 42003429435004 n  + 48792336851826 n  + 37966170733595 n

                       2                                     /
     + 18763410604137 n  + 5266457560980 n + 629318289600)  /  (512 (n + 5)
                                                           /

    (n + 4) (n + 3) (4 n + 11) (4 n + 7) (4 n + 3) (2 n + 11) (2 n + 9)

                                                      2          2          2
    (4 n + 13) (4 n + 9) (5 + 4 n) (4 n + 1) (7 + 2 n)  (5 + 2 n)  (2 n + 3)

             2
    (2 n + 1) )



                             and in Maple notation



25/512*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
2282702336*n^11+50249775456*n^10+488426050876*n^9+2761367267964*n^8+
10067671833429*n^7+24797768386797*n^6+42003429435004*n^5+48792336851826*n^4+
37966170733595*n^3+18763410604137*n^2+5266457560980*n+629318289600)/(n+5)/(n+4)
/(n+3)/(4*n+11)/(4*n+7)/(4*n+3)/(2*n+11)/(2*n+9)/(4*n+13)/(4*n+9)/(5+4*n)/(4*n+
1)/(7+2*n)^2/(5+2*n)^2/(2*n+3)^2/(2*n+1)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               11              10               9                8
875 (31800064 n   + 719836704 n   + 7222657124 n  + 42334034316 n

                     7                 6                 5                 4
     + 160768302531 n  + 414522078453 n  + 738707611556 n  + 907149783429 n

                     3                 2
     + 749313523885 n  + 394263636438 n  + 117925364700 n + 14983768800)

    (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /
    (5 n + 7) (5 n + 6)  /  (512 (n + 5) (n + 4) (n + 3) (4 n + 11) (4 n + 7)
                        /

    (4 n + 3) (2 n + 11) (2 n + 9) (4 n + 13) (4 n + 9) (5 + 4 n) (4 n + 1)

             2          2          2          2
    (7 + 2 n)  (5 + 2 n)  (2 n + 3)  (2 n + 1) )



                             and in Maple notation



875/512*(31800064*n^11+719836704*n^10+7222657124*n^9+42334034316*n^8+
160768302531*n^7+414522078453*n^6+738707611556*n^5+907149783429*n^4+
749313523885*n^3+394263636438*n^2+117925364700*n+14983768800)*(5*n+2)*(5*n+3)*(
5*n+4)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(n+5)/(n+4)/(n+3)/(4*n+
11)/(4*n+7)/(4*n+3)/(2*n+11)/(2*n+9)/(4*n+13)/(4*n+9)/(5+4*n)/(4*n+1)/(7+2*n)^2
/(5+2*n)^2/(2*n+3)^2/(2*n+1)^2










Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



          3         2
25 (1403 n  + 9651 n  + 19966 n + 11880) (5 n + 1) (5 n + 2) (5 n + 3)

    (5 n + 4) (1 + n) (5 n + 6) (5 n + 7)/(16 (n + 2) (n + 3) (n + 4) (n + 5)

    (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n) (2 n + 9) (2 n + 11))



                             and in Maple notation



25/16*(1403*n^3+9651*n^2+19966*n+11880)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(
5*n+6)*(5*n+7)/(n+2)/(n+3)/(n+4)/(n+5)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/
(2*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
75 (4195300 n  + 70833912 n  + 513139375 n  + 2083210024 n  + 5183918817 n

                   3               2
     + 8097375104 n  + 7754811928 n  + 4164421140 n + 960498000) (5 n + 1)

                                                                          /
    (5 n + 2) (5 n + 3) (5 n + 4) (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)  /  (
                                                                        /

                 2          2
    256 (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (n + 4) (n + 5)

    (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9) (2 n + 11) (4 n + 7) (4 n + 11))



                             and in Maple notation



75/256*(4195300*n^8+70833912*n^7+513139375*n^6+2083210024*n^5+5183918817*n^4+
8097375104*n^3+7754811928*n^2+4164421140*n+960498000)*(5*n+1)*(5*n+2)*(5*n+3)*(
5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)/(n
+4)/(n+5)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(2*n+11)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



             8             7              6               5               4
25 (4413452 n  + 75954192 n  + 562252869 n  + 2338976286 n  + 5982400371 n

                   3               2
     + 9636512742 n  + 9550434448 n  + 5326658040 n + 1280664000) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                 2          2
    256 (2 n + 3)  (5 + 2 n)  (2 n + 1) (n + 2) (n + 3) (n + 4) (n + 5)

    (5 + 4 n) (4 n + 9) (7 + 2 n) (2 n + 9) (2 n + 11) (4 n + 7) (4 n + 11))



                             and in Maple notation



25/256*(4413452*n^8+75954192*n^7+562252869*n^6+2338976286*n^5+5982400371*n^4+
9636512742*n^3+9550434448*n^2+5326658040*n+1280664000)*(5*n+2)*(5*n+3)*(5*n+4)*
(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(2*n+3)^2/(5+2*n)^2/(2*n+1)/(n+2)/(n+3)
/(n+4)/(n+5)/(5+4*n)/(4*n+9)/(7+2*n)/(2*n+9)/(2*n+11)/(4*n+7)/(4*n+11)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 3, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   8              7               6
    (5 n + 2) (5 n + 1) (48830432 n  + 868380564 n  + 6620480424 n

                    5                4                 3                 2
     + 28258965075 n  + 73871278878 n  + 121144869921 n  + 121789463626 n

                                      /
     + 68679163080 n + 16648632000)  /  (512 (4 n + 13) (4 n + 11) (4 n + 7)
                                    /

    (2 n + 11) (2 n + 9) (7 + 2 n) (4 n + 9) (5 + 4 n) (n + 5) (n + 4) (n + 3)

                               2          2
    (n + 2) (2 n + 1) (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



25/512*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
48830432*n^8+868380564*n^7+6620480424*n^6+28258965075*n^5+73871278878*n^4+
121144869921*n^3+121789463626*n^2+68679163080*n+16648632000)/(4*n+13)/(4*n+11)/
(4*n+7)/(2*n+11)/(2*n+9)/(7+2*n)/(4*n+9)/(5+4*n)/(n+5)/(n+4)/(n+3)/(n+2)/(2*n+1
)/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 3, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6               5               4
525 (4816768 n  + 87529836 n  + 683927316 n  + 3001666845 n  + 8095939917 n

                    3                2
     + 13747567269 n  + 14361211799 n  + 8443546650 n + 2140538400) (5 n + 2)

                                                                            /
    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (
                                                                          /

                                                                     2
    512 (7 + 2 n) (2 n + 11) (4 n + 11) (5 + 4 n) (2 n + 1) (5 + 2 n)

                                            2
    (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3)  (2 n + 9) (n + 3) (n + 4) (n + 5)

    )



                             and in Maple notation



525/512*(4816768*n^8+87529836*n^7+683927316*n^6+3001666845*n^5+8095939917*n^4+
13747567269*n^3+14361211799*n^2+8443546650*n+2140538400)*(5*n+2)*(5*n+3)*(5*n+4
)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+11)/(4*n+11)/(5+4*n)/(2*
n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*n+9)/(n+3)/(n+4)/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 3, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



             7             6              5              4               3
875 (981344 n  + 16571900 n  + 117919928 n  + 458186609 n  + 1049705078 n

                   2
     + 1417731161 n  + 1045077480 n + 324324000) (5 n + 2) (5 n + 3) (5 n + 4)

                                                                   /
    (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6)  /  (512
                                                                 /

                                                                 2
    (7 + 2 n) (2 n + 11) (4 n + 11) (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 13)

                                 2
    (4 n + 9) (4 n + 7) (2 n + 3)  (2 n + 9) (n + 3) (n + 4) (n + 5))



                             and in Maple notation



875/512*(981344*n^7+16571900*n^6+117919928*n^5+458186609*n^4+1049705078*n^3+
1417731161*n^2+1045077480*n+324324000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)
*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(7+2*n)/(2*n+11)/(4*n+11)/(5+4*n)/(2*n+1)/(5+2
*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*n+9)/(n+3)/(n+4)/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 4, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    9               8                7
    (5 n + 2) (5 n + 1) (219251432 n  + 4731770700 n  + 44650984986 n

                     6                 5                  4                  3
     + 241696282617 n  + 826815281403 n  + 1853506696635 n  + 2723191831939 n

                      2                                     /
     + 2529419632488 n  + 1348561783800 n + 314659144800)  /  (512 (4 n + 11)
                                                          /

                                                                         2
    (4 n + 7) (2 n + 11) (2 n + 9) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 3)

                                       2          2          2
    (n + 5) (n + 4) (2 n + 1) (7 + 2 n)  (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



25/512*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
219251432*n^9+4731770700*n^8+44650984986*n^7+241696282617*n^6+826815281403*n^5+
1853506696635*n^4+2723191831939*n^3+2529419632488*n^2+1348561783800*n+
314659144800)/(4*n+11)/(4*n+7)/(2*n+11)/(2*n+9)/(4*n+13)/(4*n+9)/(5+4*n)/(n+3)^
2/(n+5)/(n+4)/(2*n+1)/(7+2*n)^2/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 4, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7               6               5
2625 (957768 n  + 21083588 n  + 203434230 n  + 1129034439 n  + 3971452962 n

                   4                3                2
     + 9182424252 n  + 13957216890 n  + 13452281521 n  + 7462759950 n

     + 1816214400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 6)

                                    /
    (5 n + 7) (5 n + 8) (5 n + 9)  /  (512 (4 n + 11) (4 n + 7) (2 n + 11)
                                  /

                                                    2
    (2 n + 9) (4 n + 13) (4 n + 9) (5 + 4 n) (n + 3)  (n + 5) (n + 4) (2 n + 1)

             2          2          2
    (7 + 2 n)  (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



2625/512*(957768*n^9+21083588*n^8+203434230*n^7+1129034439*n^6+3971452962*n^5+
9182424252*n^4+13957216890*n^3+13452281521*n^2+7462759950*n+1816214400)*(5*n+2)
*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(4*n+11)/(4*n
+7)/(2*n+11)/(2*n+9)/(4*n+13)/(4*n+9)/(5+4*n)/(n+3)^2/(n+5)/(n+4)/(2*n+1)/(7+2*
n)^2/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 4, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



              9             8              7               6                5
375 (2678296 n  + 60923412 n  + 609165942 n  + 3513502335 n  + 12880606809 n

                    4                3                2
     + 31122090993 n  + 49555781123 n  + 50138987160 n  + 29245816530 n

     + 7491884400) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 11)

                                                        /               2
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (512 (7 + 2 n)
                                                      /

                                                       2
    (2 n + 11) (4 n + 11) (5 + 4 n) (2 n + 1) (5 + 2 n)  (4 n + 13) (4 n + 9)

                       2                                  2
    (4 n + 7) (2 n + 3)  (2 n + 9) (n + 5) (n + 4) (n + 3)  (n + 2))



                             and in Maple notation



375/512*(2678296*n^9+60923412*n^8+609165942*n^7+3513502335*n^6+12880606809*n^5+
31122090993*n^4+49555781123*n^3+50138987160*n^2+29245816530*n+7491884400)*(5*n+
2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(7
+2*n)^2/(2*n+11)/(4*n+11)/(5+4*n)/(2*n+1)/(5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2
*n+3)^2/(2*n+9)/(n+5)/(n+4)/(n+3)^2/(n+2)






Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 5, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4)

                                             9               8                7
    (5 n + 3) (5 n + 2) (5 n + 1) (74513336 n  + 1660865572 n  + 16181305862 n

                    6                 5                 4                  3
     + 90409954615 n  + 319203857099 n  + 738572817718 n  + 1120306040223 n

                      2                                    /
     + 1074886350075 n  + 592410766500 n + 143026884000)  /  (512 (4 n + 15)
                                                         /

    (4 n + 11) (4 n + 7) (2 n + 11) (2 n + 9) (4 n + 13) (4 n + 9) (5 + 4 n)

           2                                    2          2          2
    (n + 3)  (n + 5) (n + 4) (2 n + 1) (7 + 2 n)  (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



125/512*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*
(5*n+1)*(74513336*n^9+1660865572*n^8+16181305862*n^7+90409954615*n^6+
319203857099*n^5+738572817718*n^4+1120306040223*n^3+1074886350075*n^2+
592410766500*n+143026884000)/(4*n+15)/(4*n+11)/(4*n+7)/(2*n+11)/(2*n+9)/(4*n+13
)/(4*n+9)/(5+4*n)/(n+3)^2/(n+5)/(n+4)/(2*n+1)/(7+2*n)^2/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 5, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9             8              7               6               5
4375 (1048360 n  + 23853852 n  + 237847794 n  + 1363862901 n  + 4956426057 n

                    4                3                2
     + 11839896399 n  + 18596450849 n  + 18526397268 n  + 10627432200 n

     + 2675673000) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12) (5 n + 11)

                                                        /
    (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /  (512 (4 n + 15)
                                                      /

    (4 n + 11) (4 n + 7) (2 n + 11) (2 n + 9) (4 n + 13) (4 n + 9) (5 + 4 n)

           2                                    2          2          2
    (n + 3)  (n + 5) (n + 4) (2 n + 1) (7 + 2 n)  (5 + 2 n)  (2 n + 3) )



                             and in Maple notation



4375/512*(1048360*n^9+23853852*n^8+237847794*n^7+1363862901*n^6+4956426057*n^5+
11839896399*n^4+18596450849*n^3+18526397268*n^2+10627432200*n+2675673000)*(5*n+
2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(4
*n+15)/(4*n+11)/(4*n+7)/(2*n+11)/(2*n+9)/(4*n+13)/(4*n+9)/(5+4*n)/(n+3)^2/(n+5)
/(n+4)/(2*n+1)/(7+2*n)^2/(5+2*n)^2/(2*n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, 1 + n, n + 5, n + 2],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
125 (16225208 n  + 381287556 n  + 3937581966 n  + 23451177795 n

                    5                 4                 3                 2
     + 88759812417 n  + 221387555049 n  + 363878689969 n  + 380025365640 n

     + 228818671200 n + 60511374000) (5 n + 2) (5 n + 3) (5 n + 4) (5 n + 12)

    (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)

       /               2
      /  (512 (7 + 2 n)  (2 n + 11) (4 n + 11) (4 n + 15) (5 + 4 n) (2 n + 1)
     /

             2                                         2
    (5 + 2 n)  (4 n + 13) (4 n + 9) (4 n + 7) (2 n + 3)  (2 n + 9) (n + 5)

                   2
    (n + 4) (n + 3)  (n + 2))



                             and in Maple notation



125/512*(16225208*n^9+381287556*n^8+3937581966*n^7+23451177795*n^6+88759812417*
n^5+221387555049*n^4+363878689969*n^3+380025365640*n^2+228818671200*n+
60511374000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+6)
*(5*n+7)*(5*n+8)*(5*n+9)/(7+2*n)^2/(2*n+11)/(4*n+11)/(4*n+15)/(5+4*n)/(2*n+1)/(
5+2*n)^2/(4*n+13)/(4*n+9)/(4*n+7)/(2*n+3)^2/(2*n+9)/(n+5)/(n+4)/(n+3)^2/(n+2)








Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 2, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               9              8               7                6
125 (13550520 n  + 310828064 n  + 3104470045 n  + 17706843254 n

                    5                 4                 3                 2
     + 63508193386 n  + 148410462176 n  + 225753722521 n  + 215339625354 n

     + 116766973224 n + 27398891520) (5 n + 1) (5 n + 2) (5 n + 3) (5 n + 4)

    (1 + n) (5 n + 6) (5 n + 7) (5 n + 8)/(216 (n + 2) (n + 3) (n + 4) (n + 5)

    (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n)

    (2 n + 9) (2 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10) (3 n + 13))



                             and in Maple notation



125/216*(13550520*n^9+310828064*n^8+3104470045*n^7+17706843254*n^6+63508193386*
n^5+148410462176*n^4+225753722521*n^3+215339625354*n^2+116766973224*n+
27398891520)*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*(1+n)*(5*n+6)*(5*n+7)*(5*n+8)/(n+2
)/(n+3)/(n+4)/(n+5)/(3*n+5)/(3*n+8)/(3*n+11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2
*n+9)/(2*n+11)/(4+3*n)/(7+3*n)/(3*n+10)/(3*n+13)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 2, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                   8              7               6
    (5 n + 2) (5 n + 1) (11273289 n  + 230506308 n  + 2009557418 n

                   5                4                3                2
     + 9746281920 n  + 28735487601 n  + 52700338412 n  + 58679388892 n

     + 36254545360 n + 9513504000)/(72 (n + 2) (n + 3) (n + 4) (n + 5)

    (3 n + 5) (3 n + 8) (3 n + 11) (2 n + 1) (2 n + 3) (5 + 2 n) (7 + 2 n)

    (2 n + 9) (2 n + 11) (4 + 3 n) (7 + 3 n) (3 n + 10) (3 n + 13))



                             and in Maple notation



25/72*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
11273289*n^8+230506308*n^7+2009557418*n^6+9746281920*n^5+28735487601*n^4+
52700338412*n^3+58679388892*n^2+36254545360*n+9513504000)/(n+2)/(n+3)/(n+4)/(n+
5)/(3*n+5)/(3*n+8)/(3*n+11)/(2*n+1)/(2*n+3)/(5+2*n)/(7+2*n)/(2*n+9)/(2*n+11)/(4
+3*n)/(7+3*n)/(3*n+10)/(3*n+13)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 2, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              8             7              6              5               4
4375 (713331 n  + 14765657 n  + 130586942 n  + 644017190 n  + 1935943211 n

                   3               2
     + 3630501017 n  + 4146163268 n  + 2635458408 n + 713512800) (5 n + 2)

    (5 n + 3) (5 n + 4) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)/(432

    (n + 3) (n + 4) (n + 5) (2 n + 9) (2 n + 3) (5 + 2 n) (7 + 3 n) (2 n + 1)

    (3 n + 8) (3 n + 10) (2 n + 11) (4 + 3 n) (7 + 2 n) (3 n + 13) (3 n + 11)

    (3 n + 5))



                             and in Maple notation



4375/432*(713331*n^8+14765657*n^7+130586942*n^6+644017190*n^5+1935943211*n^4+
3630501017*n^3+4146163268*n^2+2635458408*n+713512800)*(5*n+2)*(5*n+3)*(5*n+4)*(
5*n+1)*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)/(n+3)/(n+4)/(n+5)/(2*n+9)/(2*n+3)/(5+2*n
)/(7+3*n)/(2*n+1)/(3*n+8)/(3*n+10)/(2*n+11)/(4+3*n)/(7+2*n)/(3*n+13)/(3*n+11)/(
3*n+5)




























Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 3, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    10               9                8
    (5 n + 2) (5 n + 1) (142055223 n   + 3927230196 n  + 47946546274 n

                     7                  6                  5                  4
     + 340160291356 n  + 1551777746563 n  + 4752255491824 n  + 9885549302596 n

                       3                   2
     + 13779646093984 n  + 12306556898304 n  + 6353221754880 n + 1438441804800)

       /
      /  (144 (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 11) (2 n + 9)
     /

    (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8)

                                     2
    (3 n + 5) (n + 2) (n + 3) (n + 4)  (n + 5))



                             and in Maple notation



25/144*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
142055223*n^10+3927230196*n^9+47946546274*n^8+340160291356*n^7+1551777746563*n^
6+4752255491824*n^5+9885549302596*n^4+13779646093984*n^3+12306556898304*n^2+
6353221754880*n+1438441804800)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+11)/(2*n+
9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(n+2)/(n+3
)/(n+4)^2/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 3, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



625 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                  10              9               8
    (5 n + 2) (5 n + 1) (6349515 n   + 178817829 n  + 2225924090 n

                    7                6                 5                 4
     + 16114547122 n  + 75069335139 n  + 234918183861 n  + 499647753400 n

                     3                 2                                   /
     + 712523432628 n  + 651397610016 n  + 344429624000 n + 79913433600)  /  (
                                                                         /

    48 (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 11) (2 n + 9) (7 + 2 n)

    (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5)

           2        2
    (n + 3)  (n + 4)  (n + 5))



                             and in Maple notation



625/48*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
6349515*n^10+178817829*n^9+2225924090*n^8+16114547122*n^7+75069335139*n^6+
234918183861*n^5+499647753400*n^4+712523432628*n^3+651397610016*n^2+
344429624000*n+79913433600)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+11)/(2*n+9)/
(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(n+3)^2/(n+4)
^2/(n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 3, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



              10              9               8                7
875 (5378964 n   + 152903904 n  + 1923526277 n  + 14092375186 n

                    6                 5                 4                 3
     + 66539232570 n  + 211410095956 n  + 457384285773 n  + 664805890714 n

                     2
     + 620753221216 n  + 335918103840 n + 79913433600) (5 n + 2) (5 n + 3)

                                                                             /
    (5 n + 4) (5 n + 11) (5 n + 1) (5 n + 6) (5 n + 7) (5 n + 8) (5 n + 9)  /
                                                                           /

    (144 (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 11) (2 n + 9)

    (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8)

                     2        2
    (3 n + 5) (n + 3)  (n + 4)  (n + 5))



                             and in Maple notation



875/144*(5378964*n^10+152903904*n^9+1923526277*n^8+14092375186*n^7+66539232570*
n^6+211410095956*n^5+457384285773*n^4+664805890714*n^3+620753221216*n^2+
335918103840*n+79913433600)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+11)*(5*n+1)*(5*n+6)*(5
*n+7)*(5*n+8)*(5*n+9)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+11)/(2*n+9)/(7+2*n
)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(n+3)^2/(n+4)^2/(n+
5)




























Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



25 (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4) (5 n + 3)

                                    10                9                 8
    (5 n + 2) (5 n + 1) (824671912 n   + 23135139280 n  + 286469710121 n

                      7                  6                   5
     + 2060505154902 n  + 9527449989378 n  + 29569577990760 n

                       4                   3                   2
     + 62335830785309 n  + 88067100968978 n  + 79735786719760 n

                                           /             2        2
     + 41745308220000 n + 9589612032000)  /  (216 (n + 5)  (n + 4)  (n + 3)
                                         /

    (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 11) (2 n + 9) (7 + 2 n)

    (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5))



                             and in Maple notation



25/216*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*(5*n+1)*(
824671912*n^10+23135139280*n^9+286469710121*n^8+2060505154902*n^7+9527449989378
*n^6+29569577990760*n^5+62335830785309*n^4+88067100968978*n^3+79735786719760*n^
2+41745308220000*n+9589612032000)/(n+5)^2/(n+4)^2/(n+3)/(3*n+13)/(3*n+10)/(7+3*
n)/(4+3*n)/(2*n+11)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(
3*n+8)/(3*n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



125 (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4)

                                             10               9
    (5 n + 3) (5 n + 2) (5 n + 1) (83582027 n   + 2388106025 n

                    8                 7                  6                  5
     + 30141786906 n  + 221150251482 n  + 1043720643843 n  + 3308092969305 n

                      4                   3                  2
     + 7125179705464 n  + 10289076718948 n  + 9525382461600 n

                                          /
     + 5100801182400 n + 1198701504000)  /  (144 (3 n + 13) (3 n + 10)
                                        /

    (7 + 3 n) (4 + 3 n) (2 n + 11) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3)

                                                               2        2
    (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 5)  (n + 4)

           2
    (n + 3) )



                             and in Maple notation



125/144*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*
(5*n+1)*(83582027*n^10+2388106025*n^9+30141786906*n^8+221150251482*n^7+
1043720643843*n^6+3308092969305*n^5+7125179705464*n^4+10289076718948*n^3+
9525382461600*n^2+5100801182400*n+1198701504000)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3
*n)/(2*n+11)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/
(3*n+5)/(n+5)^2/(n+4)^2/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 4, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10             9               8               7
4375 (2963477 n   + 85478534 n  + 1090450001 n  + 8097186774 n

                    6                 5                 4                 3
     + 38732654127 n  + 124626472386 n  + 272966211979 n  + 401548803346 n

                     2
     + 379363036576 n  + 207647470800 n + 49945896000) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8) (5 n + 7)

                /
    (5 n + 6)  /  (432 (3 n + 13) (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 11)
              /

    (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11)

                               2        2        2
    (3 n + 8) (3 n + 5) (n + 5)  (n + 4)  (n + 3) )



                             and in Maple notation



4375/432*(2963477*n^10+85478534*n^9+1090450001*n^8+8097186774*n^7+38732654127*n
^6+124626472386*n^5+272966211979*n^4+401548803346*n^3+379363036576*n^2+
207647470800*n+49945896000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)*(5*n+11)*(5*n+1)*(
5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+11)/(2*n+
9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)/(n+5)^2/(n
+4)^2/(n+3)^2




























Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, 1 + n, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



875 (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n) (5 n + 4)

                                             10               9
    (5 n + 3) (5 n + 2) (5 n + 1) (72928421 n   + 2056890820 n

                    8                 7                 6                  5
     + 25615993068 n  + 185386562206 n  + 862861695119 n  + 2696920923070 n

                      4                  3                  2
     + 5728372435392 n  + 8158410475584 n  + 7450563729920 n  + 3936844454400 n

                       /             2        2
     + 913296384000)  /  (432 (n + 5)  (n + 4)  (n + 3) (3 n + 16) (3 n + 13)
                     /

    (3 n + 10) (7 + 3 n) (4 + 3 n) (2 n + 11) (2 n + 9) (7 + 2 n) (5 + 2 n)

    (2 n + 3) (2 n + 1) (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5))



                             and in Maple notation



875/432*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)*(5*n+2)*
(5*n+1)*(72928421*n^10+2056890820*n^9+25615993068*n^8+185386562206*n^7+
862861695119*n^6+2696920923070*n^5+5728372435392*n^4+8158410475584*n^3+
7450563729920*n^2+3936844454400*n+913296384000)/(n+5)^2/(n+4)^2/(n+3)/(3*n+16)/
(3*n+13)/(3*n+10)/(7+3*n)/(4+3*n)/(2*n+11)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n
+1)/(3*n+14)/(3*n+11)/(3*n+8)/(3*n+5)




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, n + 2, n],

      divided by the number of SYT of shape [n,n,n,n,n] is



875 (5 n + 12) (5 n + 11) (5 n + 9) (5 n + 8) (5 n + 7) (5 n + 6) (1 + n)

                                                       10              9
    (5 n + 4) (5 n + 3) (5 n + 2) (5 n + 1) (29801049 n   + 855840435 n

                    8                7                 6                  5
     + 10860498822 n  + 80138108414 n  + 380491887601 n  + 1213661480675 n

                      4                  3                  2
     + 2631693752368 n  + 3827435156396 n  + 3570145575040 n  + 1927052635200 n

                       /
     + 456648192000)  /  (216 (3 n + 16) (3 n + 13) (3 n + 10) (7 + 3 n)
                     /

    (4 + 3 n) (2 n + 11) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1)

                                                     2        2        2
    (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 5)  (n + 4)  (n + 3) )



                             and in Maple notation



875/216*(5*n+12)*(5*n+11)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)*(1+n)*(5*n+4)*(5*n+3)
*(5*n+2)*(5*n+1)*(29801049*n^10+855840435*n^9+10860498822*n^8+80138108414*n^7+
380491887601*n^6+1213661480675*n^5+2631693752368*n^4+3827435156396*n^3+
3570145575040*n^2+1927052635200*n+456648192000)/(3*n+16)/(3*n+13)/(3*n+10)/(7+3
*n)/(4+3*n)/(2*n+11)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14)/(3*n+11)/
(3*n+8)/(3*n+5)/(n+5)^2/(n+4)^2/(n+3)^2




Theorem: The number of 5-rowed Truncated Standard Young Tableaux of shape,

    [n, n + 5, n + 5, n + 2, 1 + n],

      divided by the number of SYT of shape [n,n,n,n,n] is



               10              9               8                7
875 (12739354 n   + 369382365 n  + 4738010662 n  + 35382938634 n

                     6                 5                  4                  3
     + 170256553026 n  + 551186585145 n  + 1214923990478 n  + 1798919666976 n

                      2
     + 1710911731760 n  + 942841521600 n + 228324096000) (5 n + 2) (5 n + 3)

    (5 n + 4) (5 n + 12) (5 n + 13) (5 n + 11) (5 n + 1) (5 n + 9) (5 n + 8)

                          /
    (5 n + 7) (5 n + 6)  /  (216 (3 n + 16) (3 n + 13) (3 n + 10) (7 + 3 n)
                        /

    (4 + 3 n) (2 n + 11) (2 n + 9) (7 + 2 n) (5 + 2 n) (2 n + 3) (2 n + 1)

                                                     2        2        2
    (3 n + 14) (3 n + 11) (3 n + 8) (3 n + 5) (n + 5)  (n + 4)  (n + 3) )



                             and in Maple notation



875/216*(12739354*n^10+369382365*n^9+4738010662*n^8+35382938634*n^7+
170256553026*n^6+551186585145*n^5+1214923990478*n^4+1798919666976*n^3+
1710911731760*n^2+942841521600*n+228324096000)*(5*n+2)*(5*n+3)*(5*n+4)*(5*n+12)
*(5*n+13)*(5*n+11)*(5*n+1)*(5*n+9)*(5*n+8)*(5*n+7)*(5*n+6)/(3*n+16)/(3*n+13)/(3
*n+10)/(7+3*n)/(4+3*n)/(2*n+11)/(2*n+9)/(7+2*n)/(5+2*n)/(2*n+3)/(2*n+1)/(3*n+14
)/(3*n+11)/(3*n+8)/(3*n+5)/(n+5)^2/(n+4)^2/(n+3)^2




























                           -------------------------



        This ends this article that took, 559.450, seconds to produce.