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 A385498 #38 Aug 17 2025 09:53:27
%S A385498 1,6,48,408,3564,31626,283548,2560872,23255964,212101176,1941110628,
%T A385498 17815257048,163896843300,1510891524252,13952756564424,
%U A385498 129048895061208,1195191116753436,11082661017288264,102877353868090080,955912961224763232,8889969049985302464
%N A385498 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k).
%F A385498 a(n) = [x^n] 1/((1-3*x) * (1-x)^(3*n)).
%F A385498 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(3*n+k-1,k).
%F A385498 From _Vaclav Kotesovec_, Jul 30 2025: (Start)
%F A385498 Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(139*n^3 - 366*n^2 + 143*n + 132)*a(n) = (588665*n^6 - 2281011*n^5 + 2262209*n^4 + 1245939*n^3 - 3359986*n^2 + 1877400*n - 322560)*a(n-1) - 648*(2*n - 3)*(4*n - 7)*(4*n - 5)*(139*n^3 + 51*n^2 - 172*n + 48)*a(n-2).
%F A385498 a(n) ~ 2^(8*n + 1/2) / (sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
%F A385498 G.f.: g/((3-2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 13 2025
%F A385498 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A385498 G.f.: 1/(1 - x*g^3*(12-6*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 17 2025
%t A385498 Table[(81/8)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, -1/2]), {n,0,25}] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%o A385498 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n, k));
%Y A385498 Cf. A100192, A385004, A386699.
%Y A385498 Cf. A005810, A066381, A371814, A386701.
%K A385498 nonn,easy
%O A385498 0,2
%A A385498 _Seiichi Manyama_, Jul 30 2025