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 A331792 #36 Aug 27 2025 09:13:39
%S A331792 1,8,57,400,2810,19824,140497,999968,7143966,51206320,368094122,
%T A331792 2652720096,19159794004,138658606688,1005231020865,7299082678336,
%U A331792 53074479789878,386419850997552,2816685368479342,20553133273532000,150120362670452076
%N A331792 Expansion of ((1 - 4*x)/sqrt(1 - 8*x + 4*x^2) - 1)/(6*x^2).
%H A331792 Seiichi Manyama, <a href="/A331792/b331792.txt">Table of n, a(n) for n = 0..1000</a>
%F A331792 a(n) = (2/(n+2)) * A331515(n) = Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).
%F A331792 n * (n+2) * a(n) = (n+1) * (4 * (2*n+1) * a(n-1) - 4 * n * a(n-2)) for n>1.
%F A331792 a(n) ~ 2^(n + 1/2) * (2 + sqrt(3))^(n + 3/2) / (3^(3/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 26 2020
%F A331792 a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - _Seiichi Manyama_, Aug 24 2025
%F A331792 From _Seiichi Manyama_, Aug 27 2025: (Start)
%F A331792 a(n) = [x^n] (1+4*x+3*x^2)^(n+1).
%F A331792 E.g.f.: exp(4*x) * BesselI(1, 2*sqrt(3)*x) / sqrt(3), with offset 1. (End)
%t A331792 a[n_] := Sum[3^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, May 05 2021 *)
%o A331792 (PARI) N=20; x='x+O('x^N); Vec(((1-4*x)/sqrt(1-8*x+4*x^2)-1)/(6*x^2))
%o A331792 (PARI) {a(n) = sum(k=0, n, 3^k*binomial(n+1, k)*binomial(n+1, k+1))}
%Y A331792 Column 4 of A331791.
%Y A331792 Cf. A069835, A331515.
%K A331792 nonn,changed
%O A331792 0,2
%A A331792 _Seiichi Manyama_, Jan 26 2020