A217061 Expansion of e.g.f. exp(A006351(x)).
1, 1, 3, 15, 109, 1053, 12767, 186763, 3204313, 63128665, 1404963387, 34867190823, 954800951749, 28600649870133, 930325531322519, 32658109219519843, 1230609634110271921, 49545182501048868145, 2122562841050605554291, 96411483206025310956735, 4628163318874435745244445
Offset: 0
Keywords
Programs
-
Mathematica
CoefficientList[Series[4*ProductLog[-E^((x-1)/2)/2]^2/E^x,{x, 0, 15}], x]*Range[0, 15]! (* Vaclav Kotesovec, Aug 04 2014 *) -
Maxima
a(n):=(sum((m*sum((n+k-1)!*sum(1/(k-j)!*sum((2^(j-l)*(-1)^(l+j)*stirling1(n-m-l+j,j-l))/(l!*(n-m-l+j)!),l,0,j),j,0,k),k,0,n-m))/m!,m,1,n));
-
PARI
my(x='x+O('x^20)); apply(round, Vec(serlaplace(4*lambertw(-exp((x-1)/2)/2)^2 / exp(x)))) \\ Michel Marcus, Jan 27 2025
Formula
a(n) = sum(m=1..n, (sum(k=0..n-m, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(l=0..j, (2^(j-l)*(-1)^(l+j)*Stirling1(n-m-l+j,j-l))/(l!*(n-m-l+j)!)))))/(m-1)!), n>0, a(0)=1.
From Vaclav Kotesovec, Aug 04 2014: (Start)
E.g.f.: 4*LambertW(-exp((x-1)/2)/2)^2 / exp(x).
a(n) ~ sqrt(2) * n^(n-1) / (exp(n-1) * (2*log(2)-1)^(n-1/2)). (End)
Extensions
More terms from Michel Marcus, Jan 27 2025