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A317178 Total number of colors summed over all collections of nonempty multisets with a total of n objects having color set {1,...,k} for some k <= n.

%I A317178 #12 Dec 08 2023 07:11:41
%S A317178 0,1,6,34,210,1378,9726,73314,588846,5018290,45216798,429290178,
%T A317178 4281260386,44724792130,488185773782,5555082575762,65760600166734,
%U A317178 808339570702354,10299822335353654,135828836289073570,1851208197237464530,26040184250745321058
%N A317178 Total number of colors summed over all collections of nonempty multisets with a total of n objects having color set {1,...,k} for some k <= n.
%H A317178 Alois P. Heinz, <a href="/A317178/b317178.txt">Table of n, a(n) for n = 0..400</a>
%F A317178 a(n) = Sum_{k=0..n} k * A255903(n,k).
%p A317178 b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*add(d*
%p A317178       binomial(d+k-1, k-1), d=numtheory[divisors](j)), j=1..n)/n)
%p A317178     end:
%p A317178 a:= n-> add(add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k)*k, k=0..n):
%p A317178 seq(a(n), n=0..25);
%t A317178 b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k]*Sum[d*Binomial[d + k - 1, k - 1], {d, Divisors[j]}], {j, 1, n}]/n];
%t A317178 a[n_] := Sum[Sum[b[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]*k, {k, 0, n}];
%t A317178 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Dec 08 2023, after _Alois P. Heinz_ *)
%Y A317178 Cf. A255903.
%K A317178 nonn
%O A317178 0,3
%A A317178 _Alois P. Heinz_, Jul 23 2018