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 A307969 #21 May 12 2021 03:07:43
%S A307969 1,-2,2,4,-24,48,24,-464,1376,-704,-9920,41600,-55040,-201216,1266048,
%T A307969 -2628864,-3021312,37696512,-108659712,15857664,1067003904,
%U A307969 -4155138048,4378226688,27416125440,-149814263808,273526325248,569660309504,-5103546466304,13241145229312,4914079858688
%N A307969 Coefficient of x^n in 1/(n+1) * (1 - 2*x - 2*x^2)^(n+1).
%C A307969 Also coefficient of x^n in the expansion of 2/(1 + 2*x + sqrt(1 + 4*x + 12*x^2)).
%H A307969 Seiichi Manyama, <a href="/A307969/b307969.txt">Table of n, a(n) for n = 0..1000</a>
%F A307969 a(n) = Sum_{k=0..floor(n/2)} (-2)^(n-k) * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-2)^(n-k) * binomial(n,2*k) * A000108(k).
%F A307969 (n+2) * a(n) = -2 * (2*n+1) * a(n-1) - 12 * (n-1) * a(n-2).
%F A307969 a(n) = (-2)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 2, -2). - _Vaclav Kotesovec_, May 12 2021
%t A307969 a[n_] := Sum[(-2)^(n-k) * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] // Flatten (* _Amiram Eldar_, May 12 2021 *)
%t A307969 Table[(-2)^n * Hypergeometric2F1[1/2 - n/2, -n/2, 2, -2], {n, 0, 30}] (* _Vaclav Kotesovec_, May 12 2021 *)
%o A307969 (PARI) {a(n) = polcoef((1-2*x-2*x^2)^(n+1)/(n+1), n)}
%o A307969 (PARI) {a(n) = sum(k=0, n\2, (-2)^(n-k)*binomial(n, k)*binomial(n-k, k)/(k+1))}
%o A307969 (PARI) {a(n) = sum(k=0, n\2, (-2)^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1))}
%Y A307969 Column 2 of A307968.
%Y A307969 Cf. A000108, A071356, A116093.
%K A307969 sign
%O A307969 0,2
%A A307969 _Seiichi Manyama_, May 08 2019