A218691 Number of ways to paint some (possibly none or all) of the trees over all forests on n labeled nodes.

1, 2, 6, 26, 156, 1242, 12616, 158034, 2372880, 41725106, 843126624, 19277549898, 492447987136, 13907344659210, 430397513894016, 14487404695687298, 527023721684738304, 20605894357093102434, 861761850029367846400, 38387125875316048363386, 1814541564588778500135936

Offset: 0

Author

Geoffrey Critzer, Nov 04 2012

Keywords

• nonn

Links

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

Crossrefs

Cf. A101313.

Programs

Maple

T:= -LambertW(-x):
egf:= exp(T-T^2/2)^2:
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2012

Mathematica

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[(t-t^2/2)]^2,{x,0,nn}],x]

Formula

E.g.f.: exp(T(x) - T(x)^2/2)^2 where T(x) is e.g.f. for A000169.

a(n) = Sum_{m=1..n} A105599(n,m)*2^m.

a(n) ~ 2*n^(n-2)*exp(1). - Vaclav Kotesovec, Aug 16 2013