cp's OEIS Frontend

A075197 Number of partitions of n balls of n colors.

%I A075197 #33 Jan 11 2025 02:37:19
%S A075197 1,1,6,38,305,2777,28784,330262,4152852,56601345,829656124,
%T A075197 12992213830,216182349617,3804599096781,70540645679070,
%U A075197 1373192662197632,27982783451615363,595355578447896291,13193917702518844859,303931339674133588444,7263814501407389465610
%N A075197 Number of partitions of n balls of n colors.
%C A075197 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 A075197 Alois P. Heinz, <a href="/A075197/b075197.txt">Table of n, a(n) for n = 0..400</a>
%F A075197 a(n) = [x^n] Product_{k>=1} 1 / (1 - x^k)^binomial(k+n-1,n-1). - _Ilya Gutkovskiy_, May 09 2021
%e A075197 Illustration of first terms, ordered by number of parts, size of parts and smallest color of parts, etc.
%e A075197 a(1) = 1:
%e A075197   {{1}}
%e A075197 a(2) = 6 = 3+3:
%e A075197   {{1,1}},{{1,2}},{{2,2}},
%e A075197   {{1},{1}},{{1},{2}},{{2},{2}}
%e A075197 a(3) = 38 = 10+18+10:
%e A075197   {{1,1,1}},{{1,1,2}},{{1,1,3}},{{1,2,2}},{{1,2,3}},{{1,3,3}},
%e A075197   {{2,2,2}},{{2,2,3}},{{2,3,3}},{{3,3,3}},
%e A075197   {{1},{1,1}},{{1},{1,2}},{{1},{1,3}},{{1},{2,2}},{{1},{2,3}},{{1},{3,3}},
%e A075197   {{2},{1,1}},{{2},{1,2}},{{2},{1,3}},{{2},{2,2}},{{2},{2,3}},{{2},{3,3}},
%e A075197   {{3},{1,1}},{{3},{1,2}},{{3},{1,3}},{{3},{2,2}},{{3},{2,3}},{{3},{3,3}},
%e A075197   {{1},{1},{1}},{{1},{1},{2}},{{1},{1},{3}},{{1},{2},{2}},{{1},{2},{3}},{{1},{3},{3}},
%e A075197   {{2},{2},{2}},{{2},{2},{3}},{{2},{3},{3}},{{3},{3},{3}}
%p A075197 with(numtheory):
%p A075197 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
%p A075197       binomial(d+k-1, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
%p A075197     end:
%p A075197 a:= n-> A(n, n):
%p A075197 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 26 2012
%t A075197 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 A075197 Main diagonal of A075196.
%Y A075197 Cf. A001700 (n balls of one color in n unlabeled boxes).
%Y A075197 Cf. A209668 (boxes are ordered by size but not by content among a given size: order among boxes of the same size matters.),
%Y A075197 Cf. A261783 (compositions of balls of n colors: boxes are labeled)
%Y A075197 Cf. A252654 (lists instead of boxes : order of balls matter)
%Y A075197 Cf. A000262 (lists instead of boxes and all n colors are used)
%Y A075197 Cf. A255906 (the c colors used form the interval [1,c])
%Y A075197 Cf. A255951 (the n-1 colors used form the interval [1,n-1])
%Y A075197 Cf. A255942 (0/1 binary coloring)
%Y A075197 Cf. A066186 (only 1 color among n = n * p(n))
%Y A075197 Cf. A000110 (the n possible colors are used : set partitions of [n])
%Y A075197 Cf. A005651 (the n possible colors are used and order of parts of the same size matters)
%Y A075197 Cf. A000670 (the n possible colors are used and order of all parts matters)
%K A075197 nonn
%O A075197 0,3
%A A075197 _Christian G. Bower_, Sep 07 2002