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A309996 Number of forests of rooted identity trees with 2n colored nodes using exactly n colors.

%I A309996 #10 Sep 15 2022 04:02:18
%S A309996 1,1,60,10746,4191916,2894100710,3128432924009,4887094401176148,
%T A309996 10429904418286375276,29174096160751011237987,
%U A309996 103602945849963939278211780,455474137757927866858846385930,2428879210633773939611859814825540,15447942216555014401018067561180236424
%N A309996 Number of forests of rooted identity trees with 2n colored nodes using exactly n colors.
%H A309996 Alois P. Heinz, <a href="/A309996/b309996.txt">Table of n, a(n) for n = 0..188</a>
%F A309996 a(n) = A256068(2n+1,n).
%p A309996 b:= proc(n, k) option remember; `if`(n<2, n, add(b(n-j, k)*add(b(d, k)
%p A309996       *k*d*(-1)^(j/d+1), d=numtheory[divisors](j)), j=1..n-1)/(n-1))
%p A309996     end:
%p A309996 a:= n-> add(b(2*n+1, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A309996 seq(a(n), n=0..15);
%t A309996 b[n_, k_] := b[n, k] = If[n < 2, n, Sum[b[n - j, k]*Sum[b[d, k]*k*d*(-1)^(j/d+1), {d, Divisors[j]}], {j, 1, n-1}]/(n-1)];
%t A309996 a[n_] := Sum[b[2*n+1, n-i]*(-1)^i*Binomial[n, i], {i, 0, n}];
%t A309996 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Sep 15 2022, after _Alois P. Heinz_ *)
%Y A309996 Cf. A256068.
%K A309996 nonn
%O A309996 0,3
%A A309996 _Alois P. Heinz_, Aug 26 2019