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A309994 Number of forests of rooted trees with 2n colored nodes using exactly n colors.

%I A309994 #14 Sep 15 2022 04:08:32
%S A309994 1,2,89,14845,5613775,3809941836,4073969863427,6316651717425358,
%T A309994 13407079935176225215,37344967651943608528498,
%U A309994 132181958309965092862822183,579566807739313784043087337938,3083812115454145185391757131500066,19577110356940490275990571617295644659
%N A309994 Number of forests of rooted trees with 2n colored nodes using exactly n colors.
%H A309994 Alois P. Heinz, <a href="/A309994/b309994.txt">Table of n, a(n) for n = 0..187</a>
%F A309994 a(n) = A256064(2*n+1,n).
%p A309994 b:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*b(d, k),
%p A309994       d=numtheory[divisors](j))*b(n-j, k)*k, j=1..n-1))/(n-1))
%p A309994     end:
%p A309994 a:= n-> add(b(2*n+1, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A309994 seq(a(n), n=0..15);
%t A309994 b[n_, k_] := b[n, k] = If[n < 2, n, Sum[Sum[d*b[d, k], {d, Divisors[j]}]*b[n-j, k]*k, {j, 1, n-1}]/(n-1)];
%t A309994 a[n_] := Sum[b[2*n+1, n-i]*(-1)^i*Binomial[n, i], {i, 0, n}];
%t A309994 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Sep 15 2022, after _Alois P. Heinz_ *)
%Y A309994 Cf. A256064.
%K A309994 nonn
%O A309994 0,2
%A A309994 _Alois P. Heinz_, Aug 26 2019