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 A165984 Number of ways to put n indistinguishable balls into n^3 distinguishable boxes. 0
 1, 1, 36, 3654, 766480, 275234400, 151111164204, 117774526188844, 123672890985095232, 168324948170849366820, 288216356245328994082600, 606320062786763763996747618, 1537230010624231669678572481296, 4622745700243196227504110670860680 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS See A165817 for the case n indistinguishable balls into 2*n distinguishable boxes. See A054688 for the case n indistinguishable balls into n^2 distinguishable boxes. a(n) is the number of (weak) compositions of n into n^3 parts. - Joerg Arndt, Oct 04 2017 LINKS FORMULA a(n) = binomial(n^3+n-1, n). Let denote P(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n. Then one has: a(n) = Sum_{i=1..P(n)} (n^3)!/((n^3-p(i))!*(Product_{j=1..d(i)} m(i,j)!)). a(n) = [x^n] 1/(1 - x)^(n^3). - Ilya Gutkovskiy, Oct 03 2017 EXAMPLE For n = 2 the a(2) = 36 solutions are [0, 0, 0, 0, 0, 0, 0, 2] [0, 0, 0, 0, 0, 0, 1, 1] [0, 0, 0, 0, 0, 0, 2, 0] [0, 0, 0, 0, 0, 1, 0, 1] [0, 0, 0, 0, 0, 1, 1, 0] [0, 0, 0, 0, 0, 2, 0, 0] [0, 0, 0, 0, 1, 0, 0, 1] [0, 0, 0, 0, 1, 0, 1, 0] [0, 0, 0, 0, 1, 1, 0, 0] [0, 0, 0, 0, 2, 0, 0, 0] [0, 0, 0, 1, 0, 0, 0, 1] [0, 0, 0, 1, 0, 0, 1, 0] [0, 0, 0, 1, 0, 1, 0, 0] [0, 0, 0, 1, 1, 0, 0, 0] [0, 0, 0, 2, 0, 0, 0, 0] [0, 0, 1, 0, 0, 0, 0, 1] [0, 0, 1, 0, 0, 0, 1, 0] [0, 0, 1, 0, 0, 1, 0, 0] [0, 0, 1, 0, 1, 0, 0, 0] [0, 0, 1, 1, 0, 0, 0, 0] [0, 0, 2, 0, 0, 0, 0, 0] [0, 1, 0, 0, 0, 0, 0, 1] [0, 1, 0, 0, 0, 0, 1, 0] [0, 1, 0, 0, 0, 1, 0, 0] [0, 1, 0, 0, 1, 0, 0, 0] [0, 1, 0, 1, 0, 0, 0, 0] [0, 1, 1, 0, 0, 0, 0, 0] [0, 2, 0, 0, 0, 0, 0, 0] [1, 0, 0, 0, 0, 0, 0, 1] [1, 0, 0, 0, 0, 0, 1, 0] [1, 0, 0, 0, 0, 1, 0, 0] [1, 0, 0, 0, 1, 0, 0, 0] [1, 0, 0, 1, 0, 0, 0, 0] [1, 0, 1, 0, 0, 0, 0, 0] [1, 1, 0, 0, 0, 0, 0, 0] [2, 0, 0, 0, 0, 0, 0, 0] MAPLE a:= n-> binomial(n^3+n-1, n): seq(a(n), n=0..16); MATHEMATICA Table[Binomial[n^3 + n - 1, n], {n, 0, 13}] (* Michael De Vlieger, Oct 05 2017 *) PROG (PARI) a(n) = binomial(n^3+n-1, n); \\ Altug Alkan, Oct 03 2017 CROSSREFS Cf. A001700, A054688, A060690, A165817. Sequence in context: A120349 A120359 A194611 * A003744 A163034 A184270 Adjacent sequences:  A165981 A165982 A165983 * A165985 A165986 A165987 KEYWORD nonn AUTHOR Thomas Wieder, Oct 03 2009 STATUS approved

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