A216534 - OEIS

%I #19 Sep 05 2018 04:32:46

%S 0,1,7,62,696,9564,156115,2957072,63822024,1547005920,41624943383,

%T 1231358443776,39728327494064,1388514386058240,52264389341358675,

%U 2108028274231109632,90708364554174003184,4147927057963872055296,200876745049904503019271,10270802025081264529408000,552906921706607979733097736

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

%H Alois P. Heinz, <a href="/A216534/b216534.txt">Table of n, a(n) for n = 0..150</a>

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

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

%F 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

%p T:= -LambertW(-x):

%p a:= n-> n! *coeff(series(exp(T)*log(1/(1-T))/(1-T), x, n+1), x, n):

%p seq (a(n), n=0..20);  # _Alois P. Heinz_, Sep 08 2012

%t 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]

%t CoefficientList[Series[Log[1+LambertW[-x]]/(x*(1+1/LambertW[-x])), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 30 2013 *)

%o (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

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Sep 08 2012