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

0,4

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Antidiagonal sums of A255903.

Sequence in context: A294790 A234643 A089846 * A131868 A339294 A272064

Adjacent sequences: A258447 A258448 A258449 * A258451 A258452 A258453

Keyword

nonn

Author

Alois P. Heinz, May 30 2015

Status

approved