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 A387428 #24 Sep 04 2025 03:07:27
%S A387428 1,6,56,576,6216,68976,779456,8921856,103098976,1200177216,
%T A387428 14053176576,165337030656,1952904166656,23143984805376,
%U A387428 275062611081216,3277130498482176,39128126836409856,468065027383059456,5608576020851019776,67305503084237193216,808786974964809035776
%N A387428 Expansion of 1/sqrt(1 - 12*x - 4*x^2).
%H A387428 Vincenzo Librandi, <a href="/A387428/b387428.txt">Table of n, a(n) for n = 0..250</a>
%F A387428 a(n) = Sum_{k=0..n} (3-i)^k * (3+i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
%F A387428 a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(2*(n-k),n-k) * binomial(n-k,k).
%F A387428 n*a(n) = 6*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2) for n > 1.
%F A387428 a(n) = Sum_{k=0..floor(n/2)} 10^k * 6^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
%F A387428 a(n) = [x^n] (1+6*x+10*x^2)^n.
%F A387428 E.g.f.: exp(6*x) * BesselI(0, 2*sqrt(10)*x).
%F A387428 a(n) ~ 2^(n - 3/4) * (3 + sqrt(10))^(n + 1/2) / (5^(1/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Aug 29 2025
%t A387428 CoefficientList[Series[1/Sqrt[1-12*x-4*x^2],{x,0,20}],x] (* _Vincenzo Librandi_, Sep 04 2025 *)
%o A387428 (PARI) a(n) = sum(k=0, n\2, 10^k*6^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));
%o A387428 (Magma) m:=30; R<x>:=LaurentSeriesRing(RationalField(), m); Coefficients(R!(1/Sqrt(1 - 12*x - 4*x^2))); // _Vincenzo Librandi_, Sep 04 2025
%Y A387428 Column k=3 of A386621.
%K A387428 nonn,new
%O A387428 0,2
%A A387428 _Seiichi Manyama_, Aug 29 2025