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 A380641 #12 Jan 29 2025 08:10:33
%S A380641 1,1,19,379,8857,244801,7904251,292980619,12257946289,570627408097,
%T A380641 29212843607011,1629314013114811,98250285167099209,
%U A380641 6365331315043185889,440712959779710869707,32460639303987670526731,2533396174719346231613281,208776665140069914314618689
%N A380641 Expansion of e.g.f. exp(x/(1 - 3*x)^3).
%F A380641 a(n) = n! * Sum_{k=0..n} 3^k * binomial(3*n-2*k-1,k)/(n-k)!.
%F A380641 E.g.f.: exp( Sum_{k>=1} k*(k+1)/2 * 3^(k-1) * x^k ).
%F A380641 From _Vaclav Kotesovec_, Jan 29 2025: (Start)
%F A380641 a(n) = (12*n-11)*a(n-1) - 6*(n-1)*(9*n-19)*a(n-2) + 108*(n-3)*(n-2)*(n-1)*a(n-3) - 81*(n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
%F A380641 a(n) ~ 3^n * n^(n - 1/8) / (2 * exp(n - 4*n^(3/4)/3 - sqrt(n)/6 + n^(1/4)/72 + 1/81)) * (1 + 16957/(207360*n^(1/4))). (End)
%t A380641 Table[n! * Sum[3^k * Binomial[3*n-2*k-1,k]/(n-k)!, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Jan 29 2025 *)
%o A380641 (PARI) a(n) = n!*sum(k=0, n, 3^k*binomial(3*n-2*k-1, k)/(n-k)!);
%Y A380641 Cf. A091695, A380637.
%K A380641 nonn,easy
%O A380641 0,3
%A A380641 _Seiichi Manyama_, Jan 28 2025