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A273661 Number of forests of labeled rooted trees of height at most 1, with 2n labels, n of which are used for root nodes and any root may contain >= 1 labels.

%I A273661 #8 Feb 27 2017 06:25:04
%S A273661 1,2,30,1040,59850,5020092,568136184,82506827832,14838761544750,
%T A273661 3218688299529560,824939949711312292,245760625104930199992,
%U A273661 83971523217039191918912,32541316683315808775379000,14168363320559065768499122200,6874922021593176730438764171840
%N A273661 Number of forests of labeled rooted trees of height at most 1, with 2n labels, n of which are used for root nodes and any root may contain >= 1 labels.
%H A273661 Alois P. Heinz, <a href="/A273661/b273661.txt">Table of n, a(n) for n = 0..252</a>
%F A273661 a(n) = (2n)!/n! * [x^n] Sum_{j=0..n} Stirling2(n,j)*exp(x)^j.
%F A273661 a(n) = C(2*n,n) * Sum_{j=0..n} Stirling2(2*n,j) * j^n.
%F A273661 a(n) = A143396(2n,n).
%p A273661 a:= n-> binomial(2*n,n)*add(Stirling2(n,j)*j^n, j=0..n):
%p A273661 seq(a(n), n=0..20);
%t A273661 a[0] = 1; a[n_] := Binomial[2*n, n]*Sum[StirlingS2[n, j]*j^n, {j, 0, n}]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Feb 27 2017, translated from Maple *)
%Y A273661 Cf. A143396.
%K A273661 nonn
%O A273661 0,2
%A A273661 _Alois P. Heinz_, May 27 2016