A075197 Number of partitions of n balls of n colors.

1, 1, 6, 38, 305, 2777, 28784, 330262, 4152852, 56601345, 829656124, 12992213830, 216182349617, 3804599096781, 70540645679070, 1373192662197632, 27982783451615363, 595355578447896291, 13193917702518844859, 303931339674133588444, 7263814501407389465610

Offset: 0

Author

Christian G. Bower, Sep 07 2002

Keywords

• nonn

Comments

For each integer partition of n, consider each part of size k to be a box containing k balls of up to n color. Order among parts and especially among parts of the same size does not matter. - Olivier Gérard, Aug 26 2016

Examples

			Illustration of first terms, ordered by number of parts, size of parts and smallest color of parts, etc.
a(1) = 1:
  {{1}}
a(2) = 6 = 3+3:
  {{1,1}},{{1,2}},{{2,2}},
  {{1},{1}},{{1},{2}},{{2},{2}}
a(3) = 38 = 10+18+10:
  {{1,1,1}},{{1,1,2}},{{1,1,3}},{{1,2,2}},{{1,2,3}},{{1,3,3}},
  {{2,2,2}},{{2,2,3}},{{2,3,3}},{{3,3,3}},
  {{1},{1,1}},{{1},{1,2}},{{1},{1,3}},{{1},{2,2}},{{1},{2,3}},{{1},{3,3}},
  {{2},{1,1}},{{2},{1,2}},{{2},{1,3}},{{2},{2,2}},{{2},{2,3}},{{2},{3,3}},
  {{3},{1,1}},{{3},{1,2}},{{3},{1,3}},{{3},{2,2}},{{3},{2,3}},{{3},{3,3}},
  {{1},{1},{1}},{{1},{1},{2}},{{1},{1},{3}},{{1},{2},{2}},{{1},{2},{3}},{{1},{3},{3}},
  {{2},{2},{2}},{{2},{2},{3}},{{2},{3},{3}},{{3},{3},{3}}
		

Links

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

Crossrefs

Main diagonal of A075196.

Cf. A001700 (n balls of one color in n unlabeled boxes).

Cf. A209668 (boxes are ordered by size but not by content among a given size: order among boxes of the same size matters.),

Cf. A261783 (compositions of balls of n colors: boxes are labeled)

Cf. A252654 (lists instead of boxes : order of balls matter)

Cf. A000262 (lists instead of boxes and all n colors are used)

Cf. A255906 (the c colors used form the interval [1,c])

Cf. A255951 (the n-1 colors used form the interval [1,n-1])

Cf. A255942 (0/1 binary coloring)

Cf. A066186 (only 1 color among n = n * p(n))

Cf. A000110 (the n possible colors are used : set partitions of [n])

Cf. A005651 (the n possible colors are used and order of parts of the same size matters)

Cf. A000670 (the n possible colors are used and order of all parts matters)

Programs

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-1, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 26 2012

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*Binomial[d+k-1, k-1], {d, Divisors[j]}]*A[n-j, k], {j, 1, n}]/n]; a[n_] := A[n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Formula

a(n) = [x^n] Product_{k>=1} 1 / (1 - x^k)^binomial(k+n-1,n-1). - Ilya Gutkovskiy, May 09 2021