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.
1, 1, 19, 703, 39313, 2959921, 280935811, 32221238239, 4336213980673, 670088514363553, 116959281939738451, 22759439305951039231, 4885844614853182749649, 1147088485458553806981073, 292394958982688921734424323, 80420728320326634679448511391
Offset: 0
Keywords
Programs
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PARI
a(n) = if(n==0, 1, 3^(n-1)*(n-1)!*pollaguerre(n-1, 2*n+1, -1/3));
Formula
E.g.f.: exp( (G(3*x)-1)/3 ), where G(x) is described above.
a(n) = (n-1)! * Sum_{k=0..n-1} 3^k * binomial(3*n,k)/(n-k-1)! for n > 0.
a(n+1) = 3^n * n! * LaguerreL(n, 2*n+3, -1/3).
a(n) ~ 3^(4*n - 1/2) * n^(n-1) / (2^(2*n + 3/2) * exp(n - 1/6)). - Vaclav Kotesovec, Jan 29 2025
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
E.g.f.: exp( Series_Reversion( x/(1+3*x)^3 ) ). - Seiichi Manyama, Mar 16 2025