A159297 Number of 3D matrices with positive integer entries such that sum of all entries equals n

1, 4, 10, 25, 58, 130, 286, 620, 1329, 2827, 5977, 12559, 26227, 54493, 112849, 233272, 481616, 992955, 2043238, 4194649, 8591014, 17559133, 35833948, 73054885, 148849186, 303171755, 617306563, 1256452642, 2555937826

Offset

1,2

Comments

Equivalently, number of quadruples (i, j, k; P) such that i, j and k are positive integers and P is a composition of n into ijk parts. (A composition of n with m parts is an ordered list of m positive integers that sum to n. The number of compositions of n into m parts is given by the binomial coefficient C(n - 1, m - 1).) [Joel B. Lewis, May 07 2009]

Links

Table of n, a(n) for n=1..29.

Formula

a(n) = sum(C(n - 1, ijk - 1)) where the sum is over all triples (i, j, k) such that 0 < i, j, k and ijk <= n. [Joel B. Lewis, May 07 2009]

Example

For n=3, the 10 possible matrices are: 3 (1*1*1); (1,2) as three different vectors (1*1*2, 1*2*1, 2*1*1); (2,1) as three different vectors (1*1*2, 1*2*1, 2*1*1); and (1,1,1) as three different vectors (1*1*3, 1*3*1, 3*1*1). [Typo corrected by Joel B. Lewis, Apr 04 2011]

Mathematica

Table[Sum[Sum[Sum[Binomial[n - 1, i*j*k - 1], {i, 1, n}], {j, 1, n}], {k, 1, n}], {n, 1, 40}] (* Joel B. Lewis, May 07 2009 *)

Crossrefs

Cf. A101509

Sequence in context: A229916 A113412 A227712 * A248731 A279101 A276599

Adjacent sequences: A159294 A159295 A159296 * A159298 A159299 A159300

Keyword

nonn

Author

Lior Manor, Apr 09 2009

Extensions

More terms from Joel B. Lewis, May 07 2009

Status

approved