A318364 Expansion of e.g.f. exp(x*exp(x)/(1 - x)).

1, 1, 5, 28, 197, 1676, 16597, 186796, 2350105, 32634928, 495207881, 8144456684, 144204493765, 2733218222944, 55188182951917, 1182163846918156, 26765995313355953, 638508459302742464, 16002492517241163793, 420279349847440766284, 11540406000681962458141, 330624627443307824367616

Offset: 0

Author

Ilya Gutkovskiy, Aug 24 2018

Keywords

• nonn

Crossrefs

Cf. A000248, A000262, A000522, A007526, A318365.

Programs

Maple

a:=series(exp(x*exp(x)/(1 - x)), x=0, 22): seq(n!*coeff(a, x, n), n=0..21); # Paolo P. Lava, Mar 26 2019

Mathematica

nmax = 21; CoefficientList[Series[Exp[x Exp[x]/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Floor[Exp[1] k! - 1] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]

PARI

x = 'x + O('x^25); Vec(serlaplace(exp(x*exp(x)/(1 - x)))) \\ Michel Marcus, Aug 25 2018

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A007526(k)*binomial(n-1,k-1)*a(n-k).

a(n) ~ exp(1/4 - 3*exp(1)/2 + 2*exp(1/2)*sqrt(n) - n) * n^(n - 1/4) / sqrt(2). - Vaclav Kotesovec, Aug 25 2018