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 A380516 #26 Mar 15 2025 09:43:39
%S A380516 1,1,9,157,4129,146001,6502681,349790029,22069858497,1598577634369,
%T A380516 130757736096361,11922399644742621,1199121973234651489,
%U A380516 131887738425602277457,15748194681225620534649,2028885239529647188594381,280525944581514367875035521,41434950383158772951280658689
%N A380516 Expansion of e.g.f. exp(x*G(x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%H A380516 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A380516 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A380516 E.g.f.: exp(G(x)-1), where G(x) is described above.
%F A380516 a(n) = (n-1)! * Sum_{k=0..n-1} binomial(4*n,k)/(n-k-1)! for n > 0.
%F A380516 a(n+1) = n! * LaguerreL(n, 3*n+4, -1).
%F A380516 a(n) = (-1)^(n+1)*U(1-n, 2+3*n, -1), where U is the Tricomi confluent hypergeometric function. - _Stefano Spezia_, Jan 26 2025
%F A380516 a(n) ~ 2^(8*n + 1) * n^(n-1) / (exp(n - 1/3) * 3^(3*n + 3/2)). - _Vaclav Kotesovec_, Jan 26 2025
%F A380516 E.g.f.: exp( Series_Reversion( x/(1+x)^4 ) ). - _Seiichi Manyama_, Mar 15 2025
%t A380516 Join[{1}, Table[(n-1)! * LaguerreL[n-1, 3*n+1, -1], {n, 1, 20}]] (* _Vaclav Kotesovec_, Jan 26 2025 *)
%o A380516 (PARI) a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 3*n+1, -1));
%Y A380516 Cf. A002293, A380513, A380514, A380515.
%Y A380516 Cf. A000262, A251568, A380512.
%Y A380516 Cf. A382034.
%K A380516 nonn
%O A380516 0,3
%A A380516 _Seiichi Manyama_, Jan 26 2025