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 A331793 #36 Aug 26 2025 08:36:39
%S A331793 1,10,87,740,6285,53550,458115,3934600,33913881,293244050,2542684463,
%T A331793 22101612780,192530903461,1680415209270,14692052109915,
%U A331793 128653303453200,1128147127156785,9905115333850650,87066787614156807,766127762539955700,6747880819438628541
%N A331793 Expansion of ((1 - 5*x)/sqrt(1 - 10*x + 9*x^2) - 1)/(8*x^2).
%H A331793 Seiichi Manyama, <a href="/A331793/b331793.txt">Table of n, a(n) for n = 0..1000</a>
%F A331793 a(n) = (2/(n+2)) * A331516(n) = Sum_{k=0..n} 4^k * binomial(n+1,k) * binomial(n+1,k+1).
%F A331793 n * (n+2) * a(n) = (n+1) * (5 * (2*n+1) * a(n-1) - 9 * n * a(n-2)) for n>1.
%F A331793 a(n) ~ 3^(2*n + 3) / (2^(5/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 26 2020
%F A331793 From _Seiichi Manyama_, Aug 23 2025: (Start)
%F A331793 a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A331793 a(n) = Sum_{k=0..n} 2^k * binomial(n+1,k+1) * binomial(2*k+2,k+2). (End)
%F A331793 From _Seiichi Manyama_, Aug 25 2025: (Start)
%F A331793 a(n) = [x^n] (1+5*x+4*x^2)^(n+1).
%F A331793 E.g.f.: exp(5*x) * BesselI(1, 4*x) / 2, with offset 1. (End)
%t A331793 a[n_] := Sum[4^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, May 05 2021 *)
%o A331793 (PARI) my(N=30, x='x+O('x^N)); Vec(((1-5*x)/sqrt(1-10*x+9*x^2)-1)/(8*x^2))
%o A331793 (PARI) a(n) = sum(k=0, n, 4^k*binomial(n+1, k)*binomial(n+1, k+1));
%Y A331793 Column 5 of A331791.
%Y A331793 Cf. A331516.
%K A331793 nonn,changed
%O A331793 0,2
%A A331793 _Seiichi Manyama_, Jan 26 2020