A216534 Number of cycles in all partial functions on {1,2,...,n}.

0, 1, 7, 62, 696, 9564, 156115, 2957072, 63822024, 1547005920, 41624943383, 1231358443776, 39728327494064, 1388514386058240, 52264389341358675, 2108028274231109632, 90708364554174003184, 4147927057963872055296, 200876745049904503019271, 10270802025081264529408000, 552906921706607979733097736

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Formula

E.g.f.: exp(T(x))*log(1/(1-T(x)))/(1-T(x)), where T(x) is the e.g.f. for A000169.

a(n) = Sum_{k>0} A216520(n,k)*k.

a(n) ~ exp(1)/2 * n^n * log(n) * (1 + (gamma+log(2))/log(n) - 2*sqrt(2*Pi)/(3*sqrt(n)*log(n))), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Sep 30 2013

Maple

T:= -LambertW(-x):
a:= n-> n! *coeff(series(exp(T)*log(1/(1-T))/(1-T), x, n+1), x, n):
seq (a(n), n=0..20);  # Alois P. Heinz, Sep 08 2012

Mathematica

nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; a=Exp[t]/(1-t)^y; b=D[a, y]/.y->1; Range[0, nn]!CoefficientList[Series[b, {x, 0, nn}], x]
CoefficientList[Series[Log[1+LambertW[-x]]/(x*(1+1/LambertW[-x])), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

Prog

(PARI) x='x+O('x^30); concat([0], Vec(serlaplace( log(1+lambertw(-x))/( x*(1+ 1/lambertw(-x))) ))) \\ G. C. Greubel, Sep 04 2018

Crossrefs

Sequence in context: A289212 A060005 A055066 * A167550 A304895 A161201

Adjacent sequences: A216531 A216532 A216533 * A216535 A216536 A216537

Keyword

nonn

Author

Geoffrey Critzer, Sep 08 2012

Status

approved