A052855 Number of forests of rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

1, 1, 3, 8, 24, 71, 224, 710, 2318, 7659, 25703, 87153, 298574, 1031104, 3587263, 12558652, 44214807, 156438309, 555973965, 1983817178, 7104313970, 25525304569, 91986529421, 332408847422, 1204259931815, 4373027942634, 15914143511582, 58030451159889

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

Euler transform of A036249 (as well as first differences thereof). - Franklin T. Adams-Watters, Feb 08 2006

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1717
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 823

Crossrefs

First differences of A036249 and A029856.

Programs

Maple

spec := [S,{B=Sequence(Z,1 <= card),S=Set(C),C=Prod(B,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

max = 26; A[] = 1; Do[A[x] = Exp[Sum[A[x^k]/(1 - x^k)*x^k/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; CoefficientList[A[x] + O[x]^max, x] (* Jean-François Alcover, May 25 2018 *)

PARI

{a(n)=my(A=1+x);for(i=1,n,A=exp(sum(m=1,n,subst(A/(1-x),x,x^m+x*O(x^n))*x^m/m)));polcoeff(A,n)} /* Paul D. Hanna, Oct 26 2011 */

Formula

G.f. satisfies A(x) = exp( Sum_{n>=1} A(x^n)/(1-x^n) * x^n/n ). - Paul D. Hanna, Oct 26 2011

G.f.: A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{j>=1} Product_{k>=0} (1-x^(j+k))^a(k). - Seiichi Manyama, Jun 07 2023

Extensions

More terms from Franklin T. Adams-Watters, Feb 08 2006