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

%I A200850 #18 Apr 11 2022 18:02:42
%S A200850 1,1,1,4,13,91,511,5146,41329,544573,5704381,93001096,1203040741,
%T A200850 23391560479,360416247283,8142893840446,145661102170081,
%U A200850 3750604005834361,76415186203927129,2209120481052933868,50510327090854792861,1620053085929867956291
%N A200850 The number of forests of labeled rooted strictly binary trees (each vertex has exactly two children or none) on n nodes.
%H A200850 Alois P. Heinz, <a href="/A200850/b200850.txt">Table of n, a(n) for n = 0..200</a>
%F A200850 E.g.f.: exp(A(x)) where A(x) is the e.g.f. for A036770.
%F A200850 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
%F A200850 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
%p A200850 a:= proc(n) option remember;
%p A200850       `if`(n=0, 1, add((n-1)!/(n-1-2*j)! *binomial(2*j+1, j)/
%p A200850                        (2^j) *a(n-1-2*j), j=0..(n-1)/2))
%p A200850     end:
%p A200850 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 23 2011
%t A200850 Range[0,19]! CoefficientList[Series[Exp[(1-(1-2x^2)^(1/2))/x],{x,0,19}],x]
%Y A200850 Cf. A036770.
%K A200850 nonn
%O A200850 0,4
%A A200850 _Geoffrey Critzer_, Nov 23 2011