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A319220 Number of rooted identity trees with n colored non-root nodes where the set of colors equals {1,...,k} for some k <= n.

%I A319220 #8 Jun 02 2022 10:06:11
%S A319220 1,1,4,32,362,5454,102469,2312418,60994931,1842667249,62760237328,
%T A319220 2379922607427,99460696044565,4542324964768755,225087388544097949,
%U A319220 12029089158757401655,689679033455762592599,42228989406791157626917,2750301966874829159250696
%N A319220 Number of rooted identity trees with n colored non-root nodes where the set of colors equals {1,...,k} for some k <= n.
%H A319220 Alois P. Heinz, <a href="/A319220/b319220.txt">Table of n, a(n) for n = 0..368</a>
%F A319220 a(n) = Sum_{k=0..n} A256068(n+1,k).
%p A319220 b:= proc(n, k) option remember; `if`(n<2, n, add(b(n-j, k)*add(b(d, k)
%p A319220       *k*d*(-1)^(j/d+1), d=numtheory[divisors](j)), j=1..n-1)/(n-1))
%p A319220     end:
%p A319220 a:= n-> add(add(b(n+1, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p A319220 seq(a(n), n=0..20);
%t A319220 b[n_, k_] := b[n, k] = If[n < 2, n, Sum[b[n - j, k]*Sum[b[d, k]*
%t A319220      k*d*(-1)^(j/d + 1), {d, Divisors[j]}], {j, 1, n - 1}]/(n - 1)];
%t A319220 a[n_] := Sum[Sum[b[n+1, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t A319220 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 02 2022, after _Alois P. Heinz_ *)
%Y A319220 Cf. A256068.
%K A319220 nonn
%O A319220 0,3
%A A319220 _Alois P. Heinz_, Sep 13 2018