A361710 a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n,k)*binomial(n-1,k)^2.

0, 1, -1, -8, 15, 126, -280, -2400, 5775, 50050, -126126, -1100736, 2858856, 25069968, -66512160, -585307008, 1577585295, 13919870250, -37978905250, -335813478000, 925166131890, 8194328596740, -22754499243840, -201822515032320, 564121960420200, 5009403008531376

Offset

0,4

Comments

Conjecture: the supercongruence a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) holds for all primes p >= 5 and positive integers n and k.

Compare with A005258(n-1) = Sum_{k = 0..n-1} (-1)^k*binomial(-n,k)*binomial(n-1,k)^2.

Links

Table of n, a(n) for n=0..25.

Wikipedia, Dixon's identity.

Formula

a(n) = (1/n^2) * Sum_{k = 0..n} (-1)^(n+k) * k^2 * binomial(n,k)^3 for n >= 1.

a(n) = (1/(3*n)) * Sum_{k = 0..n} (-1)^(n+k+1) * (n - 3*k) * binomial(n,k)^3 for n >= 1.

a(2*n) = (-1)^n * (1/6) * (3*n)!/n!^3 for n >= 1; a(2*n+1) = (-1)^n * (3*n+1)/(2*n+1) * (3*n)!/n!^3.

a(2*n) = (1/3)*A361716(2*n); a(2*n+1) = A361711(2*n+1) = A361716(2*n+1).

a(2*n) = (1/6)*A245086(2*n) = (1/6)*(-1)^n*A006480(n) for n >= 1.

a(n) = hypergeom([-n, 1 - n, 1 - n], [1, 1], 1);

P-recursive: n^2*(n - 1)*(6*n^2 - 16*n + 11)*a(n) = - 6*(n - 1)*(3*n^2 - 6*n + 2)*a(n-1) - (3*n - 4)*(3*n - 5)*(3*n - 6)*(6*n^2 - 4*n + 1)*a(n-2) with a(0) = 0 and a(1) = 1.

a(n) = Sum_{k = 0..n-1} (-1)^k*binomial(n,k)*binomial(n+k-1,k)*binomial(2*n-k-1,n). - Peter Bala, Jul 01 2023

Maple

seq( add((-1)^k*binomial(n, k)*binomial(n-1, k)^2, k = 0..n-1), n = 0..25);

Prog

(PARI) a(n) = sum(k = 0, n-1, (-1)^k*binomial(n, k)*binomial(n-1, k)^2); \\ Michel Marcus, Mar 26 2023

Crossrefs

Cf. A005258, A006480, A245086, A361711, A361716.

Sequence in context: A110459 A132374 A346314 * A234534 A067686 A283821

Adjacent sequences: A361707 A361708 A361709 * A361711 A361712 A361713

Keyword

sign,easy

Author

Peter Bala, Mar 21 2023

Status

approved