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 A385004 #45 Aug 17 2025 09:53:43
%S A385004 1,5,31,200,1311,8665,57556,383556,2561871,17140007,114819351,
%T A385004 769925568,5166845124,34696155564,233113911208,1566926561740,
%U A385004 10536427052463,70872688450083,476854924775869,3209222876463192,21602639249766951,145444151677134153,979397744169608784
%N A385004 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k).
%F A385004 a(n) = [x^n] 1/((1-3*x) * (1-x)^(2*n)).
%F A385004 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k-1,k).
%F A385004 From _Vaclav Kotesovec_, Jul 30 2025: (Start)
%F A385004 Recurrence: 8*n*(2*n - 1)*(15*n - 23)*a(n) = 6*(540*n^3 - 1503*n^2 + 1239*n - 320)*a(n-1) - 81*(3*n - 5)*(3*n - 4)*(15*n - 8)*a(n-2).
%F A385004 a(n) ~ 3^(3*n) / 2^(2*n+1) * (1 + 5/(3*sqrt(3*Pi*n))). (End)
%F A385004 G.f.: g/(3-2*g)^2 where g = 1+x*g^3 is the g.f. of A001764. - _Seiichi Manyama_, Aug 13 2025
%F A385004 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A385004 G.f.: 1/(1 - x*g^2*(9-4*g)) where g = 1+x*g^3 is the g.f. of A001764. - _Seiichi Manyama_, Aug 17 2025
%t A385004 Table[(27/4)^n - Binomial[3*n, n] * (-1 + Hypergeometric2F1[1, -2*n, 1 + n, -1/2]), {n, 0, 25}] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%o A385004 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n, k));
%Y A385004 Cf. A000244, A100192, A385498, A386699.
%Y A385004 Cf. A005809, A066380, A165817, A371813, A386700.
%Y A385004 Cf. A001764, A386617.
%K A385004 nonn,easy
%O A385004 0,2
%A A385004 _Seiichi Manyama_, Jul 30 2025