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.
1, 1, 9, 157, 4129, 146001, 6502681, 349790029, 22069858497, 1598577634369, 130757736096361, 11922399644742621, 1199121973234651489, 131887738425602277457, 15748194681225620534649, 2028885239529647188594381, 280525944581514367875035521, 41434950383158772951280658689
Offset: 0
Keywords
Programs
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Mathematica
Join[{1}, Table[(n-1)! * LaguerreL[n-1, 3*n+1, -1], {n, 1, 20}]] (* Vaclav Kotesovec, Jan 26 2025 *) -
PARI
a(n) = if(n==0, 1, (n-1)!*pollaguerre(n-1, 3*n+1, -1));
Formula
E.g.f.: exp(G(x)-1), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} binomial(4*n,k)/(n-k-1)! for n > 0.
a(n+1) = n! * LaguerreL(n, 3*n+4, -1).
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
a(n) ~ 2^(8*n + 1) * n^(n-1) / (exp(n - 1/3) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Jan 26 2025
E.g.f.: exp( Series_Reversion( x/(1+x)^4 ) ). - Seiichi Manyama, Mar 15 2025