A331513 a(n) = Sum_{k=0..n} (-n)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).

1, 4, -6, 32, -170, -228, 43764, -1498880, 43826598, -1249865260, 35978752876, -1053020066976, 31153402105852, -914722450924436, 25562930671296360, -604802562457466880, 5868775340572918534, 684246820455046681380, -78372285809430441261828

Offset: 0

Author

Seiichi Manyama, Jan 19 2020

Keywords

• sign

Links

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

Crossrefs

Cf. A331511, A331512.

Programs

Mathematica

a[n_] := Sum[If[n == n-k == 0, 1, (-n)^(n-k)] * (n+k+1) * Binomial[n, k] * Binomial[n + k, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, May 05 2021 *)

PARI

{a(n) = sum(k=0, n, (-n)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}

PARI

{a(n) = polcoef((1+n*x)/(1+2*(n-2)*x+(n*x)^2)^(3/2), n)}

PARI

{a(n) = sum(k=0, n, (-n+1)^k*(k+1)*binomial(n+1, k+1)^2)}

Formula

a(n) = [x^n] (1 + n*x)/(1 + 2*(n-2)*x + (n*x)^2)^(3/2).

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