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 A371774 #14 Apr 08 2024 18:49:29
%S A371774 1,4,21,121,727,4473,27949,176549,1124332,7205511,46411744,300183757,
%T A371774 1948255421,12681654613,82755728730,541213820732,3546268982757,
%U A371774 23276100962571,153004515241866,1007131032951572,6637396253259291,43791520333601111
%N A371774 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-k+1,n-3*k).
%F A371774 a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(2*n)).
%F A371774 a(n) = binomial(1+3*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [-1-3*n, 1+n, 3/2+n], 27/4). - _Stefano Spezia_, Apr 06 2024
%F A371774 From _Vaclav Kotesovec_, Apr 08 2024: (Start)
%F A371774 Recurrence: 2*n*(2*n - 1)*(671*n^4 - 4757*n^3 + 11743*n^2 - 11533*n + 3516)*a(n) = (44957*n^6 - 350256*n^5 + 997889*n^4 - 1236792*n^3 + 563834*n^2 + 39768*n - 60480)*a(n-1) - 10*(19459*n^6 - 156741*n^5 + 461272*n^4 - 575421*n^3 + 211099*n^2 + 106572*n - 60480)*a(n-2) + (93269*n^6 - 753150*n^5 + 2221631*n^4 - 2772678*n^3 + 999800*n^2 + 543408*n - 302400)*a(n-3) - 3*(3*n - 8)*(3*n - 7)*(671*n^4 - 2073*n^3 + 1498*n^2 + 366*n - 360)*a(n-4).
%F A371774 a(n) ~ 3^(3*n + 5/2) / (11 * sqrt(Pi*n) * 2^(2*n)). (End)
%o A371774 (PARI) a(n) = sum(k=0, n\3, binomial(3*n-k+1, n-3*k));
%Y A371774 Cf. A371773, A371775, A371776.
%Y A371774 Cf. A371754, A371770.
%Y A371774 Cf. A045721.
%K A371774 nonn
%O A371774 0,2
%A A371774 _Seiichi Manyama_, Apr 05 2024