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.

1, 2, 30, 1040, 59850, 5020092, 568136184, 82506827832, 14838761544750, 3218688299529560, 824939949711312292, 245760625104930199992, 83971523217039191918912, 32541316683315808775379000, 14168363320559065768499122200, 6874922021593176730438764171840

Offset: 0

Author

Alois P. Heinz, May 27 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..252

Crossrefs

Cf. A143396.

Programs

Maple

a:= n-> binomial(2*n,n)*add(Stirling2(n,j)*j^n, j=0..n):
seq(a(n), n=0..20);

Mathematica

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 *)

Formula

a(n) = (2n)!/n! * [x^n] Sum_{j=0..n} Stirling2(n,j)*exp(x)^j.

a(n) = C(2*n,n) * Sum_{j=0..n} Stirling2(2*n,j) * j^n.

a(n) = A143396(2n,n).