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 A075197 Number of partitions of n balls of n colors. 4
 1, 1, 6, 38, 305, 2777, 28784, 330262, 4152852, 56601345, 829656124, 12992213830, 216182349617, 3804599096781, 70540645679070, 1373192662197632, 27982783451615363, 595355578447896291, 13193917702518844859, 303931339674133588444, 7263814501407389465610 (list; graph; refs; listen; history; text; internal format)
 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) Sequence in context: A263855 A221283 A064309 * A276473 A062814 A319194 Adjacent sequences:  A075194 A075195 A075196 * A075198 A075199 A075200 KEYWORD nonn AUTHOR Christian G. Bower, Sep 07 2002 STATUS approved

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Last modified July 31 23:16 EDT 2021. Contains 346377 sequences. (Running on oeis4.)