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 A385813 #30 Aug 25 2025 17:24:43
%S A385813 1,15,156,1400,11655,92925,721140,5496300,41361255,308344025,
%T A385813 2282167272,16795140180,123030071437,897791417775,6530377362480,
%U A385813 47370038320800,342794475282915,2475479922896925,17843821672113780,128412824128709400,922775179449162501,6622378039719342615
%N A385813 Expansion of 1/((1-3*x) * (1-7*x))^(3/2).
%H A385813 Vincenzo Librandi, <a href="/A385813/b385813.txt">Table of n, a(n) for n = 0..500</a>
%F A385813 n*a(n) = (10*n+5)*a(n-1) - 21*(n+1)*a(n-2) for n > 1.
%F A385813 a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A385813 a(n) = Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385813 a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
%F A385813 a(n) = binomial(n+2,2) * A182401(n).
%F A385813 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A385813 a(n) = Sum_{k=0..n} (5/2)^k * (-21/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).
%t A385813 CoefficientList[Series[1/((1-3x)*(1-7*x))^(3/2),{x,0,33}],x] (* _Vincenzo Librandi_, Aug 25 2025 *)
%o A385813 (PARI) my(N=30, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(3/2))
%o A385813 (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 25 2025
%Y A385813 Cf. A385563, A385728.
%Y A385813 Cf. A098409, A182401.
%K A385813 nonn,changed
%O A385813 0,2
%A A385813 _Seiichi Manyama_, Aug 19 2025