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

Author

Lior Manor, Apr 09 2009

Keywords

• nonn

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]

Examples

			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]
		

Crossrefs

Cf. A101509

Programs

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 *)

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]

Extensions

More terms from Joel B. Lewis, May 07 2009