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 A385716 #56 Aug 25 2025 04:23:59
%S A385716 1,21,348,5320,78135,1120287,15805972,220445316,3047961735,
%T A385716 41857891075,571725145992,7774356136092,105324231178621,
%U A385716 1422411298153125,19157947746089520,257427540725705056,3451990965984505251,46205867184493459023,617482101788090727220,8239952016851603641320
%N A385716 Expansion of 1/((1-x) * (1-13*x))^(3/2).
%H A385716 Paolo Xausa, <a href="/A385716/b385716.txt">Table of n, a(n) for n = 0..800</a>
%F A385716 n*a(n) = (14*n+7)*a(n-1) - 13*(n+1)*a(n-2) for n > 1.
%F A385716 a(n) = (1/4)^n * Sum_{k=0..n} 13^k * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A385716 a(n) = Sum_{k=0..n} 3^k * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385716 a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385716 a(n) = binomial(n+2,2) * A386362(n).
%F A385716 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A385716 a(n) = Sum_{k=0..n} (7/2)^k * (-13/14)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
%F A385716 a(n) ~ sqrt(3*n) * 13^(n + 3/2) / (36*sqrt(Pi)). - _Vaclav Kotesovec_, Aug 21 2025
%t A385716 Module[{x}, CoefficientList[Series[1/((1-x)*(1-13*x))^(3/2), {x, 0, 25}], x]] (* _Paolo Xausa_, Aug 25 2025 *)
%o A385716 (PARI) my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-13*x))^(3/2))
%Y A385716 Cf. A331516, A385563.
%Y A385716 Cf. A340973, A386362.
%K A385716 nonn,new
%O A385716 0,2
%A A385716 _Seiichi Manyama_, Aug 19 2025