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 A373518 #17 Sep 03 2025 11:11:01
%S A373518 1,1,1,1,17,81,241,3361,32481,183457,2534561,36903681,325995121,
%T A373518 4808334961,90981786897,1126128625441,18354227120321,415821040873281,
%U A373518 6714588707173441,122710186163310337,3174234862391072721,63597591858999638161,1308604168710672673841
%N A373518 Expansion of e.g.f. exp(x/(1 - x^3)^(2/3)).
%F A373518 a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*n/3-k-1,k)/(n-3*k)!.
%F A373518 a(n) == 1 mod 16.
%F A373518 From _Vaclav Kotesovec_, Sep 03 2025: (Start)
%F A373518 Recurrence: (n-9)*(2*n - 21)*a(n) = (2*n - 21)*(5*n^4 - 90*n^3 + 515*n^2 - 1229*n + 1071)*a(n-3) - (n-6)*(n-5)*(n-4)*(n-3)*(20*n^4 - 570*n^3 + 5590*n^2 - 21846*n + 30303)*a(n-6) + (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(20*n^4 - 630*n^3 + 6940*n^2 - 31104*n + 49113)*a(n-9) - (n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(2*n - 9)*(5*n^4 - 210*n^3 + 3215*n^2 - 20971*n + 47886)*a(n-12) + (n-15)*(n-14)^2*(n - 13)^2*(n-12)*(n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(n-3)*(2*n - 9)*a(n-15).
%F A373518 a(n) ~ 2^(3/10) * 5^(-1/2) * exp(5*2^(-2/5)*n^(2/5)/3 - n) * n^(n - 3/10).
%F A373518 (End)
%t A373518 nmax = 25; CoefficientList[Series[E^(x/(1 - x^3)^(2/3)), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 03 2025 *)
%o A373518 (PARI) a(n) = n!*sum(k=0, n\3, binomial(2*n/3-k-1, k)/(n-3*k)!);
%Y A373518 Cf. A293493, A373517.
%Y A373518 Cf. A373523.
%K A373518 nonn,changed
%O A373518 0,5
%A A373518 _Seiichi Manyama_, Jun 08 2024