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 A383951 #13 Aug 27 2025 12:36:27
%S A383951 1,8,75,796,8975,104532,1242241,14967680,182172627,2234388520,
%T A383951 27572562017,341929238196,4257601409425,53196292831052,
%U A383951 666600870098895,8374220026541296,105432936550339523,1329984626694890760,16805828389675759921,212684693606424187460,2695304533342226489229
%N A383951 Expansion of 1/sqrt((1-x)^3 * (1-13*x)).
%H A383951 Vincenzo Librandi, <a href="/A383951/b383951.txt">Table of n, a(n) for n = 0..800</a>
%F A383951 n*a(n) = (14*n-6)*a(n-1) - 13*n*a(n-2) for n > 1.
%F A383951 a(n) = (1/4)^n * Sum_{k=0..n} 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
%F A383951 a(n) = Sum_{k=0..n} (-3)^k * 13^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+1,n-k).
%F A383951 a(n) = Sum_{k=0..n} 3^k * binomial(2*k,k) * binomial(n+1,n-k).
%t A383951 CoefficientList[Series[1/Sqrt[(1-x)^3*(1-13*x)],{x,0,33}],x] (* _Vincenzo Librandi_, Aug 27 2025 *)
%o A383951 (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt((1-x)^3*(1-13*x)))
%o A383951 (Magma) R<x> := PowerSeriesRing(Rationals(), 34); f := 1/Sqrt((1- x)^3 * (1-13*x)); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // _Vincenzo Librandi_, Aug 27 2025
%Y A383951 Partial sums of A340973.
%K A383951 nonn,changed
%O A383951 0,2
%A A383951 _Seiichi Manyama_, Aug 19 2025