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 A387210 #19 Aug 23 2025 07:59:37
%S A387210 1,19,307,4645,67843,969337,13643533,189953659,2622877075,35982412921,
%T A387210 491057325577,6672763735183,90347244052429,1219537191931975,
%U A387210 16418449380961891,220534056531679141,2956293832279184659,39559312793250153577,528522358385088314425,7051193680459915645903
%N A387210 Expansion of sqrt((1-x) / (1-13*x)^3).
%H A387210 Vincenzo Librandi, <a href="/A387210/b387210.txt">Table of n, a(n) for n = 0..300</a>
%F A387210 n*a(n) = (14*n+5)*a(n-1) - 13*(n-1)*a(n-2) for n > 1.
%F A387210 a(n) = (1/4)^n * Sum_{k=0..n} 13^k * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A387210 a(n) = Sum_{k=0..n} 3^k * (2*k+1) * binomial(2*k,k) * binomial(n,n-k).
%F A387210 a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n,n-k).
%F A387210 a(n) ~ 4 * sqrt(3*n) * 13^(n - 1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Aug 23 2025
%t A387210 CoefficientList[Series[Sqrt[(1-x)/(1-13*x)^3],{x,0,33}],x] (* _Vincenzo Librandi_, Aug 23 2025 *)
%o A387210 (PARI) my(N=20, x='x+O('x^N)); Vec(sqrt((1-x)/(1-13*x)^3))
%o A387210 (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := Sqrt(((1-x) / (1-13*x)^3)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 23 2025
%Y A387210 Cf. A163869, A163872, A387208.
%Y A387210 Cf. A385716.
%K A387210 nonn,new
%O A387210 0,2
%A A387210 _Seiichi Manyama_, Aug 22 2025