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Number of partitions of n balls of n colors.
4

%I #28 May 10 2021 04:04:13

%S 1,1,6,38,305,2777,28784,330262,4152852,56601345,829656124,

%T 12992213830,216182349617,3804599096781,70540645679070,

%U 1373192662197632,27982783451615363,595355578447896291,13193917702518844859,303931339674133588444,7263814501407389465610

%N Number of partitions of n balls of n colors.

%C 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

%H Alois P. Heinz, <a href="/A075197/b075197.txt">Table of n, a(n) for n = 0..400</a>

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

%e Illustration of first terms, ordered by number of parts, size of parts and smallest color of parts, etc.

%e :

%e a(1)=1 :

%e {{1}}:

%e a(2)=6 = 3+3:

%e {{1,1}},{{1,2}},{{2,2}},

%e {{1},{1}},{{1},{2}},{{2},{2}}

%e :

%e a(3)=38 = 10+18+10:

%e {{1,1,1}},{{1,1,2}},{{1,1,3}},{{1,2,2}},{{1,2,3}},{{1,3,3}},

%e {{2,2,2}},{{2,2,3}},{{2,3,3}},{{3,3,3}},

%e {{1},{1,1}},{{1},{1,2}},{{1},{1,3}},{{1},{2,2}},{{1},{2,3}},{{1},{3,3}},

%e {{2},{1,1}},{{2},{1,2}},{{2},{1,3}},{{2},{2,2}},{{2},{2,3}},{{2},{3,3}},

%e {{3},{1,1}},{{3},{1,2}},{{3},{1,3}},{{3},{2,2}},{{3},{2,3}},{{3},{3,3}},

%e {{1},{1},{1}},{{1},{1},{2}},{{1},{1},{3}},{{1},{2},{2}},{{1},{2},{3}},{{1},{3},{3}},

%e {{2},{2},{2}},{{2},{2},{3}},{{2},{3},{3}},{{3},{3},{3}}}}

%p with(numtheory):

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*

%p binomial(d+k-1, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)

%p end:

%p a:= n-> A(n, n):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 26 2012

%t 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_ *)

%Y Main diagonal of A075196.

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

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

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

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

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

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

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

%Y Cf. A255942 (0/1 binary coloring)

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

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

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

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

%K nonn

%O 0,3

%A _Christian G. Bower_, Sep 07 2002