This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382527 #16 Apr 07 2025 17:48:09
%S A382527 1,252,52920,12640320,3632428800,1264085222400,529085049292800,
%T A382527 263564384219136000,154550100069421056000,105562401683780321280000,
%U A382527 83178863857362412339200000,74951718050379657373286400000,76628603945744083606044672000000,88258468221509704910254374912000000
%N A382527 a(n) = Sum_{j = 1..n} (-1)^(n+j) * j^(2*n+4) * binomial(2*n, n-j).
%C A382527 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).
%H A382527 Paolo Xausa, <a href="/A382527/b382527.txt">Table of n, a(n) for n = 1..200</a>
%H A382527 John Campbell, <a href="https://doi.org/10.5206/mt.v5i1.16724">Applications of Gosper’s nonlocal derangement identity</a>, Maple Transactions, 5, 1, Article 16724, Feb. 2025.
%F A382527 a(n) = n*(n+1)*(n+2)*(2*n+3)*(5*n-1)*(2*n+1)!/6!.
%F A382527 a(n) = (2*n)!/2 * (Sum_{1 <= i <= j <= n} i^2*j^2) = (2*n)!/2 * A060493(n).
%F A382527 a(n) = 2*n*(n+2)*(2*n+3)*(5*n-1)/((n-1)*(5*n-6)) * a(n-1) with a(1) = 1.
%p A382527 seq(add((-1)^(n+j) * j^(2*n+4) * binomial(2*n, n-j), j = 1..n), n = 1..20);
%t A382527 A382527[n_] := n*(5*n - 1)*(2*n + 4)!/2880; Array[A382527, 15] (* _Paolo Xausa_, Apr 03 2025 *)
%Y A382527 Cf. A002674, A060493.
%K A382527 nonn,easy
%O A382527 1,2
%A A382527 _Peter Bala_, Mar 30 2025