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

0,4

Links

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

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).

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]

Prog

(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 */

Crossrefs

Cf. A221864.

Sequence in context: A201338 A362355 A099021 * A136229 A138419 A366341

Adjacent sequences: A220687 A220688 A220689 * A220691 A220692 A220693

Keyword

nonn

Author

Geoffrey Critzer, Apr 13 2013

Status

approved