OFFSET
0,3
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
FORMULA
a(n) = [x^n] Product_{k>=1} 1 / (1 - x^k)^binomial(k+n-1,n-1). - Ilya Gutkovskiy, May 09 2021
EXAMPLE
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}}}}
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 *)
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)
KEYWORD
nonn
AUTHOR
Christian G. Bower, Sep 07 2002
STATUS
approved