A220690 Number of acyclic graphs on {1,2,...,n} such that the node with label 1 is in the same connected component (tree) as the node with label 2.

0, 0, 1, 4, 24, 198, 2110, 27768, 436656, 8003950, 167779068, 3961727820, 104102329504, 3013887239454, 95338047836520, 3272043459321328, 121106541865151040, 4808924948167249302, 203931444227955436816, 9198925314402386788500, 439809753701222702598528

Offset: 0

Author

Geoffrey Critzer, Apr 13 2013

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A221864.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j) *(j+1)^(j-1) *b(n-1-j), j=0..n-1))
    end:
a:= n-> add(binomial(n-2, k)*(k+2)^k*b(n-k-2), k=0..n-2):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 13 2013

Mathematica

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

PARI

N = 66;  x = 'x + O('x^N);
U = sum(n=1,N,n^(n-2)*x^n/n!);
egf = intformal(intformal( deriv(deriv(U)) * exp(U) ));
gf = serlaplace(egf) + 'c0;
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Apr 13 2013 */

Formula

E.g.f.: Double integral of U''(x)*exp(U(x)) dx^2 where U(x) is the e.g.f. for A000272.

a(n) = Sum_{k=0..n-2} binomial(n-2,k)*(k+2)^k*A001858(n-k-2).