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 A387315 #14 Aug 28 2025 12:09:31
%S A387315 1,35,825,16415,297220,5067972,82893720,1315073760,20381376015,
%T A387315 310101196405,4648184007467,68817616687365,1008344472704660,
%U A387315 14644604899082620,211073938188085620,3022082811670829676,43017189132931007655,609159438493806780405,8586490781973282553375
%N A387315 Expansion of 1/((1-x) * (1-13*x))^(5/2).
%H A387315 Vincenzo Librandi, <a href="/A387315/b387315.txt">Table of n, a(n) for n = 0..800</a>
%F A387315 n*a(n) = (14*n+21)*a(n-1) - 13*(n+3)*a(n-2) for n > 1.
%F A387315 a(n) = (-1)^n * Sum_{k=0..n} 13^k * binomial(-5/2,k) * binomial(-5/2,n-k).
%F A387315 a(n) = Sum_{k=0..n} (-12)^k * binomial(-5/2,k) * binomial(n+4,n-k).
%F A387315 a(n) = Sum_{k=0..n} 12^k * 13^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
%F A387315 a(n) = (binomial(n+4,2)/6) * A387310(n).
%F A387315 a(n) = (-1)^n * Sum_{k=0..n} 14^k * (13/14)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).
%t A387315 CoefficientList[Series[1/((1-x)*(1-13*x))^(5/2),{x,0,33}],x] (* _Vincenzo Librandi_, Aug 28 2025 *)
%o A387315 (PARI) my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-13*x))^(5/2))
%o A387315 (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-13*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 28 2025
%Y A387315 Cf. A385716, A387316.
%Y A387315 Cf. A387230.
%K A387315 nonn,new
%O A387315 0,2
%A A387315 _Seiichi Manyama_, Aug 25 2025