A382527 a(n) = Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+4) * binomial(2*n, n-j).

1, 252, 52920, 12640320, 3632428800, 1264085222400, 529085049292800, 263564384219136000, 154550100069421056000, 105562401683780321280000, 83178863857362412339200000, 74951718050379657373286400000, 76628603945744083606044672000000, 88258468221509704910254374912000000

Offset: 1

Author

Peter Bala, Mar 30 2025

Keywords

• nonn
• easy

Comments

Compare with the identities Sum_{j = 1..n} (-1)^(n+j) * j^n * binomial(n, n-j) = n! and Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+2) * binomial(2*n, n-j) = n*(n+1)*(2*n+1)/6 * (2*n)!/2 = A000330(n) * (2*n)!/2. (Campbell, Eq. 17).

Links

• Paolo Xausa, Table of n, a(n) for n = 1..200
• John Campbell, Applications of Gosper’s nonlocal derangement identity, Maple Transactions, 5, 1, Article 16724, Feb. 2025.

Crossrefs

Cf. A002674, A060493.

Programs

Maple

seq(add((-1)^(n+j) * j^(2*n+4) * binomial(2*n, n-j), j = 1..n), n = 1..20);

Mathematica

A382527[n_] := n*(5*n - 1)*(2*n + 4)!/2880; Array[A382527, 15] (* Paolo Xausa, Apr 03 2025 *)

Formula

a(n) = n*(n+1)*(n+2)*(2*n+3)*(5*n-1)*(2*n+1)!/6!.

a(n) = (2*n)!/2 * (Sum_{1 <= i <= j <= n} i^2*j^2) = (2*n)!/2 * A060493(n).

a(n) = 2*n*(n+2)*(2*n+3)*(5*n-1)/((n-1)*(5*n-6)) * a(n-1) with a(1) = 1.