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 A371814 #29 Aug 17 2025 09:53:34
%S A371814 1,2,16,128,1068,9142,79612,701864,6244892,55962920,504375396,
%T A371814 4567003520,41513817444,378596616452,3462411408136,31742042431048,
%U A371814 291616814436124,2684123914512280,24746511514749280,228491677484832896,2112549277665243328
%N A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).
%F A371814 a(n) = [x^n] 1/((1+x) * (1-x)^(3*n)).
%F A371814 a(n) = binomial(4*n-1, n)*hypergeom([1, -n], [1-4*n], -1). - _Stefano Spezia_, Apr 07 2024
%F A371814 From _Vaclav Kotesovec_, Apr 07 2024: (Start)
%F A371814 Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(415*n^3 - 1898*n^2 + 2871*n - 1436)*a(n) = (838715*n^6 - 5099533*n^5 + 12225995*n^4 - 14652035*n^3 + 9157250*n^2 - 2799192*n + 322560)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(415*n^3 - 653*n^2 + 320*n - 48)*a(n-2).
%F A371814 a(n) ~ 2^(8*n + 1/2) / (5 * sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
%F A371814 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n,k). - _Seiichi Manyama_, Jul 30 2025
%F A371814 G.f.: g/((-1+2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 13 2025
%F A371814 a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A371814 G.f.: 1/(1 - x*g^3*(-4+6*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 17 2025
%o A371814 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(4*n-k-1, n-k));
%Y A371814 Cf. A072547, A371813.
%Y A371814 Cf. A005810, A262977.
%Y A371814 Cf. A066381, A262977, A385498, A386701.
%K A371814 nonn
%O A371814 0,2
%A A371814 _Seiichi Manyama_, Apr 06 2024