These theorems are for the partial sums of the Central Binomial Coefficients for the sum from 0 to r*p-1, for f from 1 to 10 --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 1, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -1, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 2 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 3, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -3, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 3 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 9, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -9, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 4 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 29, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -29, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 5 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 99, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -99, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 6 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 351, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -351, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 7 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 1275, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -1275, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 8 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 4707, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -4707, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 9 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 17577, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -17577, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1/x + 2 + x) and for any prime p, let 10 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 66197, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -66197, mod p --------------------------------- --------------------------------- These theorems are for the partial sums of the Catalan Numbers for the sum from 0 to r*p-1, for f from 1 to 10 --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 1, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -2, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 2 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 2, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -7, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 3 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 4, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -23, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 4 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 9, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -78, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 5 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 23, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -274, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 6 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 65, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -988, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 7 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 197, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -3628, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 8 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 626, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -13495, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 9 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 2056, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -50675, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial i (1 - x) (1/x + 2 + x) and for any prime p, let 10 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 3 if p mod , 3, equals , 1, then, B(p) mod p equals , 6918, mod p if p mod , 3, equals , 2, then, B(p) mod p equals , -191673, mod p --------------------------------- --------------------------------- These theorems are for the partial sums of the Motzkin Numbers for the sum from 0 to r*p-1, for f from 1 to 10 --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 2, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -2, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 2 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 4, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -4, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 3 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 10, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -10, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 4 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 24, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -24, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 5 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 62, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -62, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 6 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 164, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -164, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 7 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 446, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -446, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 8 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 1232, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -1232, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 9 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 3446, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -3446, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial 2 i (-x + 1) (1/x + 1 + x) and for any prime p, let 10 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 4 if p mod , 4, equals , 1, then, B(p) mod p equals , 9724, mod p if p mod , 4, equals , 3, then, B(p) mod p equals , -9724, mod p --------------------------------- --------------------------------- These theorems are for the partial sums of the Central Pentagonal Coefficien\ ts for the sum from 0 to r*p-1, for f from 1 to 10 --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 if p mod , 6, equals , 1, then, B(p) mod p equals , p/3 + 2/3, mod p if p mod , 6, equals , 5, then, B(p) mod p equals , p/3 - 2/3, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 2 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 if p mod , 6, equals , 1, then, B(p) mod p equals , 2, mod p if p mod , 6, equals , 5, then, B(p) mod p equals , -2, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 3 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 if p mod , 6, equals , 1, then, B(p) mod p equals , -p + 8, mod p if p mod , 6, equals , 5, then, B(p) mod p equals , -p - 8, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 4 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 20 p if p mod , 6, equals , 1, then, B(p) mod p equals , - ---- + 98/3, mod p 3 20 p if p mod , 6, equals , 5, then, B(p) mod p equals , - ---- - 98/3, mod p 3 --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 5 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 95 p if p mod , 6, equals , 1, then, B(p) mod p equals , - ---- + 428/3, mod p 3 95 p if p mod , 6, equals , 5, then, B(p) mod p equals , - ---- - 428/3, mod p 3 --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 6 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 if p mod , 6, equals , 1, then, B(p) mod p equals , -148 p + 640, mod p if p mod , 6, equals , 5, then, B(p) mod p equals , -148 p - 640, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 7 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 2065 p if p mod , 6, equals , 1, then, B(p) mod p equals , - ------ + 8794/3, 3 mod p 2065 p if p mod , 6, equals , 5, then, B(p) mod p equals , - ------ - 8794/3, 3 mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 8 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 9680 p if p mod , 6, equals , 1, then, B(p) mod p equals , 40814/3 - ------, mod p 3 9680 p if p mod , 6, equals , 5, then, B(p) mod p equals , - 40814/3 - ------, 3 mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 9 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 if p mod , 6, equals , 1, then, B(p) mod p equals , 63768 - 15225 p, mod p if p mod , 6, equals , 5, then, B(p) mod p equals , -63768 - 15225 p, mod p --------------------------------- -------------------------------------------------------- Theorem: Define the following sequence: A(i) is the Constant term of the Lau\ rent polynomial / 1 2\i |---- + 1/x + 1 + x + x | | 2 | \ x / and for any prime p, let 10 p - 1 ----- \ B(p) = ) A(i) / ----- i = 0 then B(p) mod p equals to the following according to its remainder when d\ ivided by, 6 216800 p if p mod , 6, equals , 1, then, B(p) mod p equals , 903404/3 - --------, 3 mod p 216800 p if p mod , 6, equals , 5, then, B(p) mod p equals , - 903404/3 - --------, 3 mod p --------------------------------- --------------------------------- The whole thing took, 0.970, seconds.