A200850 The number of forests of labeled rooted strictly binary trees (each vertex has exactly two children or none) on n nodes.

1, 1, 1, 4, 13, 91, 511, 5146, 41329, 544573, 5704381, 93001096, 1203040741, 23391560479, 360416247283, 8142893840446, 145661102170081, 3750604005834361, 76415186203927129, 2209120481052933868, 50510327090854792861, 1620053085929867956291

Offset: 0

Author

Geoffrey Critzer, Nov 23 2011

Keywords

• nonn

Links

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

Crossrefs

Cf. A036770.

Programs

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, add((n-1)!/(n-1-2*j)! *binomial(2*j+1, j)/
                       (2^j) *a(n-1-2*j), j=0..(n-1)/2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 23 2011

Mathematica

Range[0,19]! CoefficientList[Series[Exp[(1-(1-2x^2)^(1/2))/x],{x,0,19}],x]

Formula

E.g.f.: exp(A(x)) where A(x) is the e.g.f. for A036770.

Recurrence: 2*a(n) = -(n-2)*(n+1)*a(n-1) + 2*(n-1)*(2*n-3)*a(n-2) + 2*(n-3)*(n-2)*(n-1)^2*a(n-3). - Vaclav Kotesovec, Aug 14 2013

a(n) ~ 2^(n/2+1/2)*n^(n-1)*exp(-n-sqrt(2))*(exp(2*sqrt(2))-(-1)^n). - Vaclav Kotesovec, Aug 14 2013