A365287 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^3*A(x)^3).
1, 1, 2, 6, 48, 720, 11520, 183960, 3185280, 65681280, 1637193600, 46436544000, 1423113753600, 46607434473600, 1648149184281600, 63369409495392000, 2634451417524326400, 117088187211284889600, 5518546983426135859200, 275022667579200532992000
Offset: 0
Keywords
Programs
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Mathematica
Join[{1},Table[n!/(n+1) * Sum[(n-3*k)^k * Binomial[n+1,n-3*k]/k!, {k,0,Floor[n/3]}], {n,1,20}]] (* Vaclav Kotesovec, Nov 08 2023 *) -
PARI
a(n) = n!*sum(k=0, n\3, (n-3*k)^k*binomial(n+1, n-3*k)/k!)/(n+1);
Formula
a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n+1,n-3*k)/k!.
a(n) ~ 3^(n/3) * (1 + 4*LambertW(3^(1/4)/4))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(3^(1/4)/4)) * 2^(8*n/3 + 4) * exp(n) * LambertW(3^(1/4)/4)^(4*n/3 + 3/2)). - Vaclav Kotesovec, Nov 08 2023