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 A373517 #19 Sep 03 2025 06:31:30
%S A373517 1,1,1,1,9,41,121,1401,11761,61489,864081,10597841,81833401,
%T A373517 1350154521,21715461769,225232218121,4267472824161,84597818284001,
%U A373517 1111699778741281,23801969674626849,558853937533757161,8943028907965939081,213696639293901810201
%N A373517 Expansion of e.g.f. exp(x/(1 - x^3)^(1/3)).
%F A373517 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n/3-1,k)/(n-3*k)!.
%F A373517 a(n) == 1 mod 8.
%F A373517 From _Vaclav Kotesovec_, Sep 03 2025: (Start)
%F A373517 a(n) = (4*n^3 - 36*n^2 + 112*n - 119)*a(n-3) - 2*(n-6)*(n-5)*(n-4)*(n-3)*(3*n^2 - 27*n + 64)*a(n-6) + 4*(n-9)*(n-8)*(n-7)*(n-6)^3*(n-5)*(n-4)*(n-3)*a(n-9) - (n-12)*(n-11)*(n-10)*(n-9)^2*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*a(n-12).
%F A373517 a(n) ~ (1/2) * exp(4*n^(1/4)/3 - n) * n^(n - 3/8) * (1 - 35/(96*n^(1/4)) - 4367/(18432*sqrt(n)) + 1737829/(5308416*n^(3/4))). (End)
%t A373517 nmax = 25; CoefficientList[Series[E^(x/(1 - x^3)^(1/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 03 2025 *)
%o A373517 (PARI) a(n) = n!*sum(k=0, n\3, binomial(n/3-1, k)/(n-3*k)!);
%Y A373517 Cf. A012150, A293493, A373518.
%Y A373517 Cf. A373522.
%K A373517 nonn,changed
%O A373517 0,5
%A A373517 _Seiichi Manyama_, Jun 08 2024