Explicit Expressions for The Number of ways of playing Stanley Solitaire sta\

    ring with position [b,0^k,a] for all a>b and k from 0 to , 15



                              By Shalosh B. Ekhad



Theorem Number, 0, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, a],  is



                              (a + b)! (a + 1 - b)
                              --------------------
                                  (a + 1)! b!



                             and in Maple notation



(a+b)!/(a+1)!/b!*(a+1-b)


        For example if a=30 and b=20 then the number is, 16723268860760



This agrees with the value obtained by numerical dynamical programming that is,

    16723268860760



Theorem Number, 1, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, a],  is



                            (a + 1 + b)! (a + 2 - b)
                            ------------------------
                                  (a + 2)! b!



                             and in Maple notation



(a+1+b)!/(a+2)!/b!*(a+2-b)


        For example if a=30 and b=20 then the number is, 29075683360185



This agrees with the value obtained by numerical dynamical programming that is,

    29075683360185



Theorem Number, 2, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, a],  is



                                        2                2
                 (a + b)! (a + 3 - b) (a  + (b + 3) a + b  + 2)
                 ----------------------------------------------
                                  (a + 3)! b!



                             and in Maple notation



(a+b)!*(a+3-b)*(a^2+(b+3)*a+b^2+2)/(a+3)!/b!


        For example if a=30 and b=20 then the number is, 37281832852810



This agrees with the value obtained by numerical dynamical programming that is,

    37281832852810



Theorem Number, 3, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, 0, a],  is



                             2    2
               (a + 4 - b) (a  + b  + 5 a - 3 b + 6) (a + 1 + b)!
               --------------------------------------------------
                                  (a + 4)! b!



                             and in Maple notation



(a+4-b)*(a^2+b^2+5*a-3*b+6)/(a+4)!*(a+1+b)!/b!


        For example if a=30 and b=20 then the number is, 42205522548385



This agrees with the value obtained by numerical dynamical programming that is,

    42205522548385



Theorem Number, 4, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, 0, 0, a],  is



           4             3     2              2     3                   4
(a + b)! (a  + (b + 10) a  + (b  + 5 b + 35) a  + (b  + 10 b + 50) a + b

          3       2
     - 5 b  + 10 b  + 24) (a + 5 - b)/((a + 5)! b!)



                             and in Maple notation



(a+b)!*(a^4+(b+10)*a^3+(b^2+5*b+35)*a^2+(b^3+10*b+50)*a+b^4-5*b^3+10*b^2+24)*(a
+5-b)/(a+5)!/b!


        For example if a=30 and b=20 then the number is, 44878382668840



This agrees with the value obtained by numerical dynamical programming that is,

    44878382668840



Theorem Number, 5, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, 0, 0, 0, a],  is



                           4       3     2              2
(a + 1 + b)! (a + 6 - b) (a  + 14 a  + (b  - 5 b + 71) a

           2                    4       3       2
     + (7 b  - 35 b + 154) a + b  - 10 b  + 41 b  - 80 b + 120)/((a + 6)! b!)



                             and in Maple notation



(a+1+b)!*(a+6-b)*(a^4+14*a^3+(b^2-5*b+71)*a^2+(7*b^2-35*b+154)*a+b^4-10*b^3+41*
b^2-80*b+120)/(a+6)!/b!


        For example if a=30 and b=20 then the number is, 46191366587660



This agrees with the value obtained by numerical dynamical programming that is,

    46191366587660



Theorem Number, 6, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, 0, 0, 0, 0, a],  is



                       6             5     2                4
(a + b)! (a + 7 - b) (a  + (b + 21) a  + (b  + 14 b + 175) a

         3      2                3     4       2                  2
     + (b  + 7 b  + 77 b + 735) a  + (b  + 28 b  + 196 b + 1624) a

         5      4       3                      6       5       4        3
     + (b  - 7 b  + 28 b  + 252 b + 1764) a + b  - 14 b  + 77 b  - 196 b

            2
     + 252 b  + 720)/((a + 7)! b!)



                             and in Maple notation



(a+b)!*(a+7-b)*(a^6+(b+21)*a^5+(b^2+14*b+175)*a^4+(b^3+7*b^2+77*b+735)*a^3+(b^4
+28*b^2+196*b+1624)*a^2+(b^5-7*b^4+28*b^3+252*b+1764)*a+b^6-14*b^5+77*b^4-196*b
^3+252*b^2+720)/(a+7)!/b!


        For example if a=30 and b=20 then the number is, 46774351725360



This agrees with the value obtained by numerical dynamical programming that is,

    46774351725360



Theorem Number, 7, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, 0, 0, 0, 0, 0, a],  is



                           6       5     2               4
(a + 8 - b) (a + 1 + b)! (a  + 27 a  + (b  - 7 b + 295) a

            2                  3     4       3        2                  2
     + (18 b  - 126 b + 1665) a  + (b  - 14 b  + 174 b  - 875 b + 5104) a

           4        3        2                       6       5        4
     + (9 b  - 126 b  + 837 b  - 2772 b + 8028) a + b  - 21 b  + 183 b

            3         2
     - 847 b  + 2304 b  - 3780 b + 5040)/((a + 8)! b!)



                             and in Maple notation



(a+8-b)*(a+1+b)!*(a^6+27*a^5+(b^2-7*b+295)*a^4+(18*b^2-126*b+1665)*a^3+(b^4-14*
b^3+174*b^2-875*b+5104)*a^2+(9*b^4-126*b^3+837*b^2-2772*b+8028)*a+b^6-21*b^5+
183*b^4-847*b^3+2304*b^2-3780*b+5040)/(a+8)!/b!


        For example if a=30 and b=20 then the number is, 47007812592860



This agrees with the value obtained by numerical dynamical programming that is,

    47007812592860



Theorem Number, 8, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, 0, 0, 0, 0, 0, 0, a],  is



                       8             7     2                6
(a + b)! (a + 9 - b) (a  + (b + 36) a  + (b  + 27 b + 546) a

         3       2                  5
     + (b  + 18 b  + 303 b + 4536) a

         4      3        2                    4
     + (b  + 9 b  + 141 b  + 1809 b + 22449) a

         5       3        2                    3
     + (b  + 60 b  + 540 b  + 6168 b + 67284) a

         6      5       4         2                      2
     + (b  - 9 b  + 60 b  + 1308 b  + 11772 b + 118124) a

         7       6        5        4         3                          8
     + (b  - 18 b  + 141 b  - 540 b  + 1308 b  + 12176 b + 109584) a + b

           7        6         5         4          3          2
     - 27 b  + 303 b  - 1809 b  + 6168 b  - 11772 b  + 12176 b  + 40320)/(

    (a + 9)! b!)



                             and in Maple notation



(a+b)!*(a+9-b)*(a^8+(b+36)*a^7+(b^2+27*b+546)*a^6+(b^3+18*b^2+303*b+4536)*a^5+(
b^4+9*b^3+141*b^2+1809*b+22449)*a^4+(b^5+60*b^3+540*b^2+6168*b+67284)*a^3+(b^6-\
9*b^5+60*b^4+1308*b^2+11772*b+118124)*a^2+(b^7-18*b^6+141*b^5-540*b^4+1308*b^3+
12176*b+109584)*a+b^8-27*b^7+303*b^6-1809*b^5+6168*b^4-11772*b^3+12176*b^2+
40320)/(a+9)!/b!


        For example if a=30 and b=20 then the number is, 47091858505160



This agrees with the value obtained by numerical dynamical programming that is,

    47091858505160



Theorem Number, 9, If b<a, then the number of ways of playing Stanley Solitai\

    re starting with position, [b, 0, 0, 0, 0, 0, 0, 0, 0, 0, a],  is



               8       7     2               6        2                  5
(a + 1 + b)! (a  + 44 a  + (b  - 9 b + 826) a  + (33 b  - 297 b + 8624) a

         4       3        2                    4
     + (b  - 18 b  + 534 b  - 4077 b + 54649) a

            4        3         2                      3
     + (22 b  - 396 b  + 5093 b  - 29799 b + 214676) a

         6       5        4         3          2                       2
     + (b  - 27 b  + 444 b  - 4347 b  + 29979 b  - 123282 b + 509004) a

            6        5         4          3           2
     + (11 b  - 297 b  + 3553 b  - 23859 b  + 102168 b  - 277992 b + 663696) a

        8       7        6         5          4          3           2
     + b  - 36 b  + 556 b  - 4806 b  + 25579 b  - 87174 b  + 194904 b

     - 290304 b + 362880) (a + 10 - b)/((a + 10)! b!)



                             and in Maple notation



(a+1+b)!*(a^8+44*a^7+(b^2-9*b+826)*a^6+(33*b^2-297*b+8624)*a^5+(b^4-18*b^3+534*
b^2-4077*b+54649)*a^4+(22*b^4-396*b^3+5093*b^2-29799*b+214676)*a^3+(b^6-27*b^5+
444*b^4-4347*b^3+29979*b^2-123282*b+509004)*a^2+(11*b^6-297*b^5+3553*b^4-23859*
b^3+102168*b^2-277992*b+663696)*a+b^8-36*b^7+556*b^6-4806*b^5+25579*b^4-87174*b
^3+194904*b^2-290304*b+362880)*(a+10-b)/(a+10)!/b!


        For example if a=30 and b=20 then the number is, 47118939965790



This agrees with the value obtained by numerical dynamical programming that is,

    47118939965790



Theorem Number, 10, If b<a, then the number of ways of playing Stanley Solita\

    ire starting with position, [b, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a],  is



                        10             9     2                 8
(a + b)! (a + 11 - b) (a   + (b + 55) a  + (b  + 44 b + 1320) a

         3       2                   7
     + (b  + 33 b  + 836 b + 18150) a

         4       3        2                     6
     + (b  + 22 b  + 473 b  + 8954 b + 157773) a

         5       4        3         2                      5
     + (b  + 11 b  + 231 b  + 3751 b  + 59279 b + 902055) a

         6        4         3          2                        4
     + (b  + 110 b  + 1210 b  + 18018 b  + 249986 b + 3416930) a

         7       6        5         3          2                        3
     + (b  - 11 b  + 110 b  + 4708 b  + 51788 b  + 667084 b + 8409500) a  +

      8       7        6         5         4          2
    (b  - 22 b  + 231 b  - 1210 b  + 4708 b  + 97416 b  + 1071576 b + 12753576)

     2     9       8        7         6          5          4          3
    a  + (b  - 33 b  + 473 b  - 3751 b  + 18018 b  - 51788 b  + 97416 b

                                 10       9        8         7          6
     + 966240 b + 10628640) a + b   - 44 b  + 836 b  - 8954 b  + 59279 b

               5           4            3           2
     - 249986 b  + 667084 b  - 1071576 b  + 966240 b  + 3628800)/((a + 11)! b!)



                             and in Maple notation



(a+b)!*(a+11-b)*(a^10+(b+55)*a^9+(b^2+44*b+1320)*a^8+(b^3+33*b^2+836*b+18150)*a
^7+(b^4+22*b^3+473*b^2+8954*b+157773)*a^6+(b^5+11*b^4+231*b^3+3751*b^2+59279*b+
902055)*a^5+(b^6+110*b^4+1210*b^3+18018*b^2+249986*b+3416930)*a^4+(b^7-11*b^6+
110*b^5+4708*b^3+51788*b^2+667084*b+8409500)*a^3+(b^8-22*b^7+231*b^6-1210*b^5+
4708*b^4+97416*b^2+1071576*b+12753576)*a^2+(b^9-33*b^8+473*b^7-3751*b^6+18018*b
^5-51788*b^4+97416*b^3+966240*b+10628640)*a+b^10-44*b^9+836*b^8-8954*b^7+59279*
b^6-249986*b^5+667084*b^4-1071576*b^3+966240*b^2+3628800)/(a+11)!/b!


        For example if a=30 and b=20 then the number is, 47126706810260



This agrees with the value obtained by numerical dynamical programming that is,

    47126706810260



Theorem Number, 11, If b<a, then the number of ways of playing Stanley Solita\

    ire starting with position, [b, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a],  is



  10       9     2                 8        2                   7
(a   + 65 a  + (b  - 11 b + 1860) a  + (52 b  - 572 b + 30810) a

         4       3         2                      6
     + (b  - 22 b  + 1293 b  - 12892 b + 326613) a

            4        3          2                        5
     + (39 b  - 858 b  + 19669 b  - 164450 b + 2310945) a

         6       5         4          3           2                          4
     + (b  - 33 b  + 1016 b  - 15697 b  + 197212 b  - 1300189 b + 11028590) a

     +

         6        5          4           3            2
    (26 b  - 858 b  + 15431 b  - 166452 b  + 1320111 b  - 6544538 b + 34967140)

     3     8       7         6          5           4            3            2
    a  + (b  - 44 b  + 1029 b  - 15323 b  + 157084 b  - 1117281 b  + 5802682 b

                               2        8        7          6           5
     - 20633228 b + 70290936) a  + (13 b  - 572 b  + 11180 b  - 126698 b

               4            3             2                               10
     + 923819 b  - 4543682 b  + 15680860 b  - 37872120 b + 80627040) a + b

           9         8          7           6            5            4
     - 55 b  + 1332 b  - 18678 b  + 167949 b  - 1014255 b  + 4198118 b

                 3             2
     - 11987492 b  + 23773080 b  - 33264000 b + 39916800) (a + 1 + b)!

    (a + 12 - b)/((a + 12)! b!)



                             and in Maple notation



(a^10+65*a^9+(b^2-11*b+1860)*a^8+(52*b^2-572*b+30810)*a^7+(b^4-22*b^3+1293*b^2-\
12892*b+326613)*a^6+(39*b^4-858*b^3+19669*b^2-164450*b+2310945)*a^5+(b^6-33*b^5
+1016*b^4-15697*b^3+197212*b^2-1300189*b+11028590)*a^4+(26*b^6-858*b^5+15431*b^
4-166452*b^3+1320111*b^2-6544538*b+34967140)*a^3+(b^8-44*b^7+1029*b^6-15323*b^5
+157084*b^4-1117281*b^3+5802682*b^2-20633228*b+70290936)*a^2+(13*b^8-572*b^7+
11180*b^6-126698*b^5+923819*b^4-4543682*b^3+15680860*b^2-37872120*b+80627040)*a
+b^10-55*b^9+1332*b^8-18678*b^7+167949*b^6-1014255*b^5+4198118*b^4-11987492*b^3
+23773080*b^2-33264000*b+39916800)*(a+1+b)!*(a+12-b)/(a+12)!/b!


        For example if a=30 and b=20 then the number is, 47128675365310



This agrees with the value obtained by numerical dynamical programming that is,

    47128675365310



Theorem Number, 12, If b<a, then the number of ways of playing Stanley Solita\

    ire starting with position, [b, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a],

     is



           12             11     2                 10
(a + b)! (a   + (b + 78) a   + (b  + 65 b + 2717) a

         3       2                    9
     + (b  + 52 b  + 1872 b + 55770) a

         4       3         2                      8
     + (b  + 39 b  + 1196 b  + 31434 b + 749463) a

         5       4        3          2                        7
     + (b  + 26 b  + 689 b  + 15886 b  + 340821 b + 6926634) a

         6       5        4         3           2                          6
     + (b  + 13 b  + 351 b  + 6929 b  + 134303 b  + 2495961 b + 44990231) a  +

      7        5         4          3           2                            5
    (b  + 182 b  + 2366 b  + 44226 b  + 750022 b  + 12542738 b + 206070150) a

         8       7        6          4           3            2
     + (b  - 13 b  + 182 b  + 13468 b  + 175084 b  + 2792452 b  + 43014556 b

                   4     9       8        7         6          5           3
     + 657206836) a  + (b  - 26 b  + 351 b  - 2366 b  + 13468 b  + 516360 b

                2                             3     10       9        8
     + 6712680 b  + 98017608 b + 1414014888) a  + (b   - 39 b  + 689 b

             7          6           5           4             2
     - 6929 b  + 44226 b  - 175084 b  + 516360 b  + 10752768 b  + 139785984 b

                    2     11       10         9          8           7
     + 1931559552) a  + (b   - 52 b   + 1196 b  - 15886 b  + 134303 b

               6            5            4             3
     - 750022 b  + 2792452 b  - 6712680 b  + 10752768 b  + 114341760 b

                        12       11         10          9           8
     + 1486442880) a + b   - 65 b   + 1872 b   - 31434 b  + 340821 b

                7             6             5             4              3
     - 2495961 b  + 12542738 b  - 43014556 b  + 98017608 b  - 139785984 b

                  2
     + 114341760 b  + 479001600) (a + 13 - b)/((a + 13)! b!)



                             and in Maple notation



(a+b)!*(a^12+(b+78)*a^11+(b^2+65*b+2717)*a^10+(b^3+52*b^2+1872*b+55770)*a^9+(b^
4+39*b^3+1196*b^2+31434*b+749463)*a^8+(b^5+26*b^4+689*b^3+15886*b^2+340821*b+
6926634)*a^7+(b^6+13*b^5+351*b^4+6929*b^3+134303*b^2+2495961*b+44990231)*a^6+(b
^7+182*b^5+2366*b^4+44226*b^3+750022*b^2+12542738*b+206070150)*a^5+(b^8-13*b^7+
182*b^6+13468*b^4+175084*b^3+2792452*b^2+43014556*b+657206836)*a^4+(b^9-26*b^8+
351*b^7-2366*b^6+13468*b^5+516360*b^3+6712680*b^2+98017608*b+1414014888)*a^3+(b
^10-39*b^9+689*b^8-6929*b^7+44226*b^6-175084*b^5+516360*b^4+10752768*b^2+
139785984*b+1931559552)*a^2+(b^11-52*b^10+1196*b^9-15886*b^8+134303*b^7-750022*
b^6+2792452*b^5-6712680*b^4+10752768*b^3+114341760*b+1486442880)*a+b^12-65*b^11
+1872*b^10-31434*b^9+340821*b^8-2495961*b^7+12542738*b^6-43014556*b^5+98017608*
b^4-139785984*b^3+114341760*b^2+479001600)*(a+13-b)/(a+13)!/b!


        For example if a=30 and b=20 then the number is, 47129112359560



This agrees with the value obtained by numerical dynamical programming that is,

    47129112359560



Theorem Number, 13, If b<a, then the number of ways of playing Stanley Solita\

    ire starting with position, [b, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a],

     is



                             10       9         8          7           6
(a + 1 + b)! (6227020800 + (b   - 65 b  + 2125 b  - 44590 b  + 653447 b

                5             4              3               2
     - 6906601 b  + 53713155 b  - 309488400 b  + 1351541808 b  - 4330359072 b

                     2        10        9          8           7            6
     + 13020978816) a  + (15 b   - 975 b  + 28500 b  - 493350 b  + 5623455 b

                 5              4               3               2
     - 44455515 b  + 250789950 b  - 1025622000 b  + 3093837120 b

                                         8       7         6          5
     - 6887350080 b + 13575738240) a + (b  - 52 b  + 1913 b  - 43849 b

               4            3             2                              4
     + 717917 b  - 8047741 b  + 70846635 b  - 407868344 b + 2664929476) a  + (

        8         7          6           5            4             3
    30 b  - 1560 b  + 40515 b  - 657345 b  + 7446885 b  - 60447855 b

                  2                               3
     + 374921550 b  - 1640237820 b + 7292774280) a

         6       5         4          3           2                           6
     + (b  - 39 b  + 2095 b  - 43485 b  + 887317 b  - 8046285 b + 119753843) a

            6         5          4           3            2
     + (45 b  - 1755 b  + 47025 b  - 728325 b  + 9405765 b  - 69085575 b

                   5        2                   9
     + 671189310) a  + (75 b  - 975 b + 87450) a

         4       3         2                       8
     + (b  - 26 b  + 2671 b  - 32526 b + 1387023) a

            4         3          2                         7
     + (60 b  - 1560 b  + 59010 b  - 635310 b + 15282630) a

         2                 10         10    12    12          9           8
     + (b  - 13 b + 3641) a   + 2731 b   + a   + b   - 56680 b  + 775867 b

                  4            7             6               3              5
     + 861057588 b  - 7376902 b  + 49998885 b  - 2183234976 b  - 244234276 b

           11               2       11
     - 78 b   + 3986504640 b  + 90 a   - 5337446400 b) (a + 14 - b)/((a + 14)!

    b!)



                             and in Maple notation



(a+1+b)!*(6227020800+(b^10-65*b^9+2125*b^8-44590*b^7+653447*b^6-6906601*b^5+
53713155*b^4-309488400*b^3+1351541808*b^2-4330359072*b+13020978816)*a^2+(15*b^
10-975*b^9+28500*b^8-493350*b^7+5623455*b^6-44455515*b^5+250789950*b^4-\
1025622000*b^3+3093837120*b^2-6887350080*b+13575738240)*a+(b^8-52*b^7+1913*b^6-\
43849*b^5+717917*b^4-8047741*b^3+70846635*b^2-407868344*b+2664929476)*a^4+(30*b
^8-1560*b^7+40515*b^6-657345*b^5+7446885*b^4-60447855*b^3+374921550*b^2-\
1640237820*b+7292774280)*a^3+(b^6-39*b^5+2095*b^4-43485*b^3+887317*b^2-8046285*
b+119753843)*a^6+(45*b^6-1755*b^5+47025*b^4-728325*b^3+9405765*b^2-69085575*b+
671189310)*a^5+(75*b^2-975*b+87450)*a^9+(b^4-26*b^3+2671*b^2-32526*b+1387023)*a
^8+(60*b^4-1560*b^3+59010*b^2-635310*b+15282630)*a^7+(b^2-13*b+3641)*a^10+2731*
b^10+a^12+b^12-56680*b^9+775867*b^8+861057588*b^4-7376902*b^7+49998885*b^6-\
2183234976*b^3-244234276*b^5-78*b^11+3986504640*b^2+90*a^11-5337446400*b)*(a+14
-b)/(a+14)!/b!


        For example if a=30 and b=20 then the number is, 47129196353260



This agrees with the value obtained by numerical dynamical programming that is,

    47129196353260



Theorem Number, 14, If b<a, then the number of ways of playing Stanley Solita\

    ire starting with position,

    [b, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a],  is



                                              10    14       13         12
(a + 15 - b) (a + b)! (87178291200 + 1422247 b   + a   - 90 b   + 3655 b

                  13     2                 12
     + (b + 105) a   + (b  + 90 b + 5005) a

         3       2                     11
     + (b  + 75 b  + 3655 b + 143325) a

         4       3         2                       10
     + (b  + 60 b  + 2530 b  + 88500 b + 2749747) a

         5       4         3          2                          9
     + (b  + 45 b  + 1630 b  + 50550 b  + 1422247 b + 37312275) a

         6       5        4          3           2                            8
     + (b  + 30 b  + 955 b  + 26100 b  + 663997 b  + 15978570 b + 368411615) a

         7       6        5          4           3            2
     + (b  + 15 b  + 505 b  + 11775 b  + 272497 b  + 6018615 b  + 128733065 b

                    7     8        6         5          4            3
     + 2681453775) a  + (b  + 280 b  + 4200 b  + 95872 b  + 1931160 b

                 2                               6     9       8        7
     + 38453840 b  + 750457800 b + 14409322928) a  + (b  - 15 b  + 280 b

              5           4            3              2
     + 32872 b  + 493080 b  + 9486440 b  + 173650200 b  + 3152455928 b

                     5     10       9        8         7          6
     + 56663366760) a  + (b   - 30 b  + 505 b  - 4200 b  + 32872 b

                4             3              2
     + 2090240 b  + 31353600 b  + 547702928 b  + 9376527840 b + 159721605680)

     4     11       10        9          8          7           6            5
    a  + (b   - 45 b   + 955 b  - 11775 b  + 95872 b  - 493080 b  + 2090240 b

                 3               2                                  3     12
     + 77398928 b  + 1160983920 b  + 19073688080 b + 310989260400) a  + (b

           11         10          9           8            7            6
     - 60 b   + 1630 b   - 26100 b  + 272497 b  - 1931160 b  + 9486440 b

                 5             4               2
     - 31353600 b  + 77398928 b  + 1658929280 b  + 24883939200 b + 392156797824

       2     13       12         11          10           9            8
    ) a  + (b   - 75 b   + 2530 b   - 50550 b   + 663997 b  - 6018615 b

                 7              6              5               4
     + 38453840 b  - 173650200 b  + 547702928 b  - 1160983920 b

                   3                                               9
     + 1658929280 b  + 18897709824 b + 283465647360) a - 15978570 b

                  8    14                4              7               6
     + 128733065 b  + b   + 19073688080 b  - 750457800 b  + 3152455928 b

                    3               5          11                2
     - 24883939200 b  - 9376527840 b  - 88500 b   + 18897709824 b )/((a + 15)!

    b!)



                             and in Maple notation



(a+15-b)*(a+b)!*(87178291200+1422247*b^10+a^14-90*b^13+3655*b^12+(b+105)*a^13+(
b^2+90*b+5005)*a^12+(b^3+75*b^2+3655*b+143325)*a^11+(b^4+60*b^3+2530*b^2+88500*
b+2749747)*a^10+(b^5+45*b^4+1630*b^3+50550*b^2+1422247*b+37312275)*a^9+(b^6+30*
b^5+955*b^4+26100*b^3+663997*b^2+15978570*b+368411615)*a^8+(b^7+15*b^6+505*b^5+
11775*b^4+272497*b^3+6018615*b^2+128733065*b+2681453775)*a^7+(b^8+280*b^6+4200*
b^5+95872*b^4+1931160*b^3+38453840*b^2+750457800*b+14409322928)*a^6+(b^9-15*b^8
+280*b^7+32872*b^5+493080*b^4+9486440*b^3+173650200*b^2+3152455928*b+
56663366760)*a^5+(b^10-30*b^9+505*b^8-4200*b^7+32872*b^6+2090240*b^4+31353600*b
^3+547702928*b^2+9376527840*b+159721605680)*a^4+(b^11-45*b^10+955*b^9-11775*b^8
+95872*b^7-493080*b^6+2090240*b^5+77398928*b^3+1160983920*b^2+19073688080*b+
310989260400)*a^3+(b^12-60*b^11+1630*b^10-26100*b^9+272497*b^8-1931160*b^7+
9486440*b^6-31353600*b^5+77398928*b^4+1658929280*b^2+24883939200*b+392156797824
)*a^2+(b^13-75*b^12+2530*b^11-50550*b^10+663997*b^9-6018615*b^8+38453840*b^7-\
173650200*b^6+547702928*b^5-1160983920*b^4+1658929280*b^3+18897709824*b+
283465647360)*a-15978570*b^9+128733065*b^8+b^14+19073688080*b^4-750457800*b^7+
3152455928*b^6-24883939200*b^3-9376527840*b^5-88500*b^11+18897709824*b^2)/(a+15
)!/b!


        For example if a=30 and b=20 then the number is, 47129210125200



This agrees with the value obtained by numerical dynamical programming that is,

    47129210125200



Theorem Number, 15, If b<a, then the number of ways of playing Stanley Solita\

    ire starting with position,

    [b, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, a],  is



                                             12       11         10           9
(a + 1 + b)! (a + 16 - b) (1307674368000 + (b   - 90 b   + 3976 b   - 112575 b

                8             7              6               5                4
     + 2249017 b  - 33083520 b  + 366199590 b  - 3085250175 b  + 19937234808 b

                    3                 2                                     2
     - 99175269240 b  + 384330316320 b  - 1138984642800 b + 3134328981120) a

            12         11          10            9             8              7
     + (17 b   - 1530 b   + 62679 b   - 1545300 b  + 25567575 b  - 299918250 b

                   6                5                4                 3
     + 2571236281 b  - 16385204520 b  + 78339925368 b  - 282526691040 b

                     2                                             10         9
     + 775681174800 b  - 1626292080000 b + 3031488633600) a + (34 b   - 2550 b

              8            7             6              5               4
     + 92565 b  - 2111400 b  + 33610904 b  - 391956930 b  + 3455049825 b

                    3                 2                                    3
     - 23280021000 b  + 123353399304 b  - 490685614560 b + 1922666722704) a  +

      10       9         8           7            6             5
    (b   - 75 b  + 3445 b  - 105450 b  + 2295992 b  - 36224640 b

                  4               3                2
     + 430951230 b  - 3863843775 b  + 27610644888 b  - 142597453320 b

                      4        8         7           6            5
     + 784146622896) a  + (51 b  - 3060 b  + 106046 b  - 2362320 b

                 4              3               2
     + 38904704 b  - 462002370 b  + 4473660449 b  - 29252496360 b

                      5     8       7         6           5            4
     + 225525484184) a  + (b  - 60 b  + 3428 b  - 107010 b  + 2548472 b

                 3              2                                6
     - 39869160 b  + 535069350 b  - 4329750375 b + 47277726496) a  +

      6       5         4           3            2                            8
    (b  - 45 b  + 3925 b  - 100875 b  + 3129817 b  - 35978505 b + 854224943) a

            6         5           4            3             2
     + (68 b  - 3060 b  + 119510 b  - 2437800 b  + 47573888 b  - 465174570 b

                    7     4       3         2                       10
     + 7353403057) a  + (b  - 30 b  + 4936 b  - 70665 b + 4687683) a

            4         3           2                          9
     + (85 b  - 2550 b  + 149345 b  - 1953300 b + 73870797) a

         2                 12         2                     11            10
     + (b  - 15 b + 6461) a   + (102 b  - 1530 b + 211939) a   + 2808483 b

        14        13        13         12             9              8    14
     + a   + 119 a   - 105 b   + 5021 b   - 38747475 b  + 391819343 b  + b

                     4               7                6                 3
     + 221184866736 b  - 2950297455 b  + 16645777216 b  - 514181332080 b

                    5           11                 2
     - 70321311720 b  - 144765 b   + 883233590400 b  - 1144215072000 b)/(

    (a + 16)! b!)



                             and in Maple notation



(a+1+b)!*(a+16-b)*(1307674368000+(b^12-90*b^11+3976*b^10-112575*b^9+2249017*b^8
-33083520*b^7+366199590*b^6-3085250175*b^5+19937234808*b^4-99175269240*b^3+
384330316320*b^2-1138984642800*b+3134328981120)*a^2+(17*b^12-1530*b^11+62679*b^
10-1545300*b^9+25567575*b^8-299918250*b^7+2571236281*b^6-16385204520*b^5+
78339925368*b^4-282526691040*b^3+775681174800*b^2-1626292080000*b+3031488633600
)*a+(34*b^10-2550*b^9+92565*b^8-2111400*b^7+33610904*b^6-391956930*b^5+
3455049825*b^4-23280021000*b^3+123353399304*b^2-490685614560*b+1922666722704)*a
^3+(b^10-75*b^9+3445*b^8-105450*b^7+2295992*b^6-36224640*b^5+430951230*b^4-\
3863843775*b^3+27610644888*b^2-142597453320*b+784146622896)*a^4+(51*b^8-3060*b^
7+106046*b^6-2362320*b^5+38904704*b^4-462002370*b^3+4473660449*b^2-29252496360*
b+225525484184)*a^5+(b^8-60*b^7+3428*b^6-107010*b^5+2548472*b^4-39869160*b^3+
535069350*b^2-4329750375*b+47277726496)*a^6+(b^6-45*b^5+3925*b^4-100875*b^3+
3129817*b^2-35978505*b+854224943)*a^8+(68*b^6-3060*b^5+119510*b^4-2437800*b^3+
47573888*b^2-465174570*b+7353403057)*a^7+(b^4-30*b^3+4936*b^2-70665*b+4687683)*
a^10+(85*b^4-2550*b^3+149345*b^2-1953300*b+73870797)*a^9+(b^2-15*b+6461)*a^12+(
102*b^2-1530*b+211939)*a^11+2808483*b^10+a^14+119*a^13-105*b^13+5021*b^12-\
38747475*b^9+391819343*b^8+b^14+221184866736*b^4-2950297455*b^7+16645777216*b^6
-514181332080*b^3-70321311720*b^5-144765*b^11+883233590400*b^2-1144215072000*b)
/(a+16)!/b!


        For example if a=30 and b=20 then the number is, 47129212013660



This agrees with the value obtained by numerical dynamical programming that is,

    47129212013660



                          ---------------------------



        This concludes this paper that took, 1.168, seconds to produce