A258450 Number of collections of nonempty multisets of colored objects, where n is the number of objects plus the number of distinct colors.

1, 0, 1, 2, 5, 13, 35, 100, 298, 926, 2995, 10045, 34871, 125040, 462283, 1759340, 6882479, 27639252, 113809750, 479993898, 2071411798, 9138568984, 41182104446, 189418562699, 888607018626, 4248949407337, 20695172225549, 102617378820155, 517728263280060

Offset: 0

Author

Alois P. Heinz, May 30 2015

Keywords

• nonn

Examples

			a(4) = 5: {{1},{1},{1}}, {{1},{1,1}}, {{1,1,1}}, {{1},{2}}, {{1,2}}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Antidiagonal sums of A255903.

Programs

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
      add(d*binomial(d+k-1, k-1), d=divisors(j)), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
a:= n-> add(T(n-i, i), i=0..n/2):
seq(a(n), n=0..30);

Mathematica

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-j, k]*DivisorSum[j, #*Binomial[# +k-1, k-1]&], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
a[n_] := Sum[T[n-i, i], {i, 0, n/2}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

a(n) = Sum_{i=0..floor(n/2)} A255903(n-i,i).