A382042 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
1, 1, 3, 37, 733, 20181, 714541, 30903769, 1579206441, 93099946249, 6219777779641, 464382363698661, 38319628830696973, 3463058939163189133, 340172205752538636933, 36087128101110502864561, 4111807211977470782285521, 500807663307856030823859729, 64931674940413564774656214513
Offset: 0
Keywords
Programs
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PARI
a(n) = if(n==0, 1, n!*sum(k=0, n-1, (3*k+1)^(n-k-1)*binomial(n+k, k)/((n+k)*(n-k-1)!)));
Formula
Let F(x) be the e.g.f. of A382039. F(x) = log(A(x))/x = C(x*A(x)^3).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(3*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (3*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.