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A380638 Expansion of e.g.f. exp(x*G(4*x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

%I A380638 #13 Jan 29 2025 07:57:05
%S A380638 1,1,33,2209,226753,31555521,5557183201,1185423664993,297171500140929,
%T A380638 85638231765516673,27896677183469054881,10137203757416219332641,
%U A380638 4065668625283435566910273,1783936343221839549449049409,850091650335726912762794748513,437197222292805469886634467693281
%N A380638 Expansion of e.g.f. exp(x*G(4*x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%H A380638 Wikipedia, <a href="https://en.wikipedia.org/wiki/Laguerre_polynomials">Laguerre polynomials</a>
%H A380638 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A380638 E.g.f.: exp( (G(4*x)-1)/4 ), where G(x) is described above.
%F A380638 a(n) = (n-1)! * Sum_{k=0..n-1} 4^k * binomial(4*n,k)/(n-k-1)! for n > 0.
%F A380638 a(n+1) = 4^n * n! * LaguerreL(n, 3*n+4, -1/4).
%F A380638 a(n) ~ 2^(10*n - 1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 1/12)). - _Vaclav Kotesovec_, Jan 29 2025
%F A380638 a(n) = (-4)^(n-1)*U(1-n, 2+3*n, -1/4), where U is the Tricomi confluent hypergeometric function. - _Stefano Spezia_, Jan 29 2025
%o A380638 (PARI) a(n) = if(n==0, 1, 4^(n-1)*(n-1)!*pollaguerre(n-1, 3*n+1, -1/4));
%Y A380638 Cf. A002293, A380516.
%K A380638 nonn
%O A380638 0,3
%A A380638 _Seiichi Manyama_, Jan 28 2025