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 A331552 #26 Sep 08 2022 08:46:25
%S A331552 1,2,-6,-12,30,60,-140,-280,630,1260,-2772,-5544,12012,24024,-51480,
%T A331552 -102960,218790,437580,-923780,-1847560,3879876,7759752,-16224936,
%U A331552 -32449872,67603900,135207800,-280816200,-561632400,1163381400,2326762800,-4808643120,-9617286240,19835652870
%N A331552 Expansion of (1 + 2*x)/(1 + 4*x^2)^(3/2).
%H A331552 Seiichi Manyama, <a href="/A331552/b331552.txt">Table of n, a(n) for n = 0..1000</a>
%F A331552 |a(n)| = A100071(n+1).
%F A331552 a(n) = Sum_{k=0..n} (-2)^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).
%F A331552 a(n) = Sum_{k=0..n} (-1)^k * (k+1) * binomial(n+1,k+1)^2.
%F A331552 n * (2*n-1) * a(n) = 2 * a(n-1) - 4 * n * (2*n+1) * a(n-2) for n>1.
%F A331552 E.g.f.: (1 + 2*x)*BesselJ(0,2*x) - 2*x*BesselJ(1,2*x). - _Ilya Gutkovskiy_, Mar 04 2021
%t A331552 a[n_] := Sum[(-1)^k * (k + 1) * Binomial[n + 1, k + 1]^2, {k, 0, n}]; Array[a, 33, 0] (* _Amiram Eldar_, Jan 20 2020 *)
%o A331552 (PARI) N=66; x='x+O('x^N); Vec((1+2*x)/(1+4*x^2)^(3/2))
%o A331552 (PARI) {a(n) = sum(k=0, n, (-2)^(n-k)*(n+k+1)*binomial(n, k)*binomial(n+k, k))}
%o A331552 (PARI) {a(n) = sum(k=0, n, (-1)^k*(k+1)*binomial(n+1, k+1)^2)}
%o A331552 (Magma) R<x>:=PowerSeriesRing(Rationals(), 33); Coefficients(R!( (1 + 2*x)/(1 + 4*x^2)^(3/2))); // _Marius A. Burtea_, Jan 20 2020
%o A331552 (Magma) [&+[(-1)^k*(k+1)*Binomial(n+1, k+1)^2:k in [0..n]]:n in [0..33]]; // _Marius A. Burtea_, Jan 20 2020
%Y A331552 Column 1 of A331511.
%Y A331552 Cf. A100071.
%K A331552 sign
%O A331552 0,2
%A A331552 _Seiichi Manyama_, Jan 20 2020