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 A380637 #19 Mar 16 2025 10:08:59
%S A380637 1,1,19,703,39313,2959921,280935811,32221238239,4336213980673,
%T A380637 670088514363553,116959281939738451,22759439305951039231,
%U A380637 4885844614853182749649,1147088485458553806981073,292394958982688921734424323,80420728320326634679448511391
%N A380637 Expansion of e.g.f. exp(x*G(3*x)^3) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
%H A380637 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A380637 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A380637 E.g.f.: exp( (G(3*x)-1)/3 ), where G(x) is described above.
%F A380637 a(n) = (n-1)! * Sum_{k=0..n-1} 3^k * binomial(3*n,k)/(n-k-1)! for n > 0.
%F A380637 a(n+1) = 3^n * n! * LaguerreL(n, 2*n+3, -1/3).
%F A380637 a(n) ~ 3^(4*n - 1/2) * n^(n-1) / (2^(2*n + 3/2) * exp(n - 1/6)). - _Vaclav Kotesovec_, Jan 29 2025
%F A380637 a(n) = (-3)^(n-1)*U(1-n, 2*(1+n), -1/3), where U is the Tricomi confluent hypergeometric function. - _Stefano Spezia_, Jan 29 2025
%F A380637 E.g.f.: exp( Series_Reversion( x/(1+3*x)^3 ) ). - _Seiichi Manyama_, Mar 16 2025
%o A380637 (PARI) a(n) = if(n==0, 1, 3^(n-1)*(n-1)!*pollaguerre(n-1, 2*n+1, -1/3));
%Y A380637 Cf. A001764, A380512.
%Y A380637 Cf. A380641.
%K A380637 nonn
%O A380637 0,3
%A A380637 _Seiichi Manyama_, Jan 28 2025