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 A331513 #18 May 05 2021 01:53:00
%S A331513 1,4,-6,32,-170,-228,43764,-1498880,43826598,-1249865260,35978752876,
%T A331513 -1053020066976,31153402105852,-914722450924436,25562930671296360,
%U A331513 -604802562457466880,5868775340572918534,684246820455046681380,-78372285809430441261828
%N A331513 a(n) = Sum_{k=0..n} (-n)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).
%H A331513 Seiichi Manyama, <a href="/A331513/b331513.txt">Table of n, a(n) for n = 0..386</a>
%F A331513 a(n) = [x^n] (1 + n*x)/(1 + 2*(n-2)*x + (n*x)^2)^(3/2).
%F A331513 a(n) = Sum_{k=0..n} (-n+1)^k * (k+1) * binomial(n+1,k+1)^2.
%t A331513 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 *)
%o A331513 (PARI) {a(n) = sum(k=0, n, (-n)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
%o A331513 (PARI) {a(n) = polcoef((1+n*x)/(1+2*(n-2)*x+(n*x)^2)^(3/2), n)}
%o A331513 (PARI) {a(n) = sum(k=0, n, (-n+1)^k*(k+1)*binomial(n+1, k+1)^2)}
%Y A331513 Cf. A331511, A331512.
%K A331513 sign
%O A331513 0,2
%A A331513 _Seiichi Manyama_, Jan 19 2020