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A165984 Number of ways to put n indistinguishable balls into n^3 distinguishable boxes. 0
1, 36, 3654, 766480, 275234400, 151111164204, 117774526188844, 123672890985095232, 168324948170849366820, 288216356245328994082600, 606320062786763763996747618, 1537230010624231669678572481296 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..12.

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.

Furthermore let sum_{i=1}^{P(n)} be a sum over i and

prod_{j=1}^{d(i)} a product over j.

Then one has:

a(n)=sum_{i=1}^{P(n)} (n^3)!/((n^3-p(i))!*(prod_{j=1}^{d(i)} m(i,j)!)).

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

for n from 1 to 16 do a[n] := binomial(n^3+n-1, n) end do;

CROSSREFS

A001700, A165817, A054688, A060690.

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

EXTENSIONS

Replaced full-length URL's for sequences by standard A-numbers - R. J. Mathar, Oct 08 2009

STATUS

approved

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Last modified November 20 21:20 EST 2014. Contains 249754 sequences.