A307911 Coefficient of x^n in expansion of (1 - n*x - n*x^2)^n.

1, -1, 0, 27, -416, 5625, -74304, 924385, -8626176, -48361131, 7124800000, -340421390199, 13686496542720, -522760216822129, 19658830846298112, -735037915447265625, 27218267709730979840, -980444996625142158435, 32830565919734078521344, -889052809376495994642527

Offset: 0

Author

Seiichi Manyama, May 05 2019

Keywords

• sign

Comments

Also coefficient of x^n in the expansion of 1/sqrt(1 + 2*n*x + n*(n+4)*x^2).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..386

Crossrefs

Main diagonal of A307819.

Cf. A092366, A307862.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[(-n)^(n-k) * Binomial[n, 2*k] * Binomial[2*k, k], {k, 0, Floor[n/2]}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Join[{1}, Table[(-n)^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -4/n], {n, 1, 20}]] (* Vaclav Kotesovec, May 12 2021 *)

PARI

{a(n) = polcoef((1-n*x-n*x^2)^n, n)}

PARI

{a(n) = sum(k=0, n\2, (-n)^(n-k)*binomial(n, k)*binomial(n-k, k))}

PARI

{a(n) = sum(k=0, n\2, (-n)^(n-k)*binomial(n, 2*k)*binomial(2*k, k))}

Formula

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

For n>0, a(n) = (-n)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 1, -4/n). - Vaclav Kotesovec, May 12 2021