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.

0, 1, 6, 34, 210, 1378, 9726, 73314, 588846, 5018290, 45216798, 429290178, 4281260386, 44724792130, 488185773782, 5555082575762, 65760600166734, 808339570702354, 10299822335353654, 135828836289073570, 1851208197237464530, 26040184250745321058

Offset: 0

Author

Alois P. Heinz, Jul 23 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A255903.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*add(d*
      binomial(d+k-1, k-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k)*k, k=0..n):
seq(a(n), n=0..25);

Mathematica

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];
a[n_] := Sum[Sum[b[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]*k, {k, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A255903(n,k).