A365283 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)^2).
1, 1, 2, 12, 120, 1380, 19440, 341040, 7029120, 164762640, 4355769600, 128527439040, 4181332700160, 148633442717760, 5734427199621120, 238676208285715200, 10659325532663808000, 508452777299622355200, 25800664274991135129600
Offset: 0
Keywords
Programs
-
Mathematica
Join[{1},Table[n!/(n+1) * Sum[(n-2*k)^k * Binomial[n+1,n-2*k]/k!, {k, 0, Floor[n/2]}], {n,1,20}]] (* Vaclav Kotesovec, Nov 08 2023 *) -
PARI
a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+1, n-2*k)/k!)/(n+1);
Formula
a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+1,n-2*k)/k!.
a(n) ~ 2^(n/2) * (1 + 3*LambertW(2^(1/3)/3))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(2^(1/3)/3)) * 3^(3*n/2 + 2) * exp(n) * LambertW(2^(1/3)/3)^(3*(n+1)/2)). - Vaclav Kotesovec, Nov 08 2023