A159297 - OEIS

%I #11 Aug 25 2019 05:42:40

%S 1,4,10,25,58,130,286,620,1329,2827,5977,12559,26227,54493,112849,

%T 233272,481616,992955,2043238,4194649,8591014,17559133,35833948,

%U 73054885,148849186,303171755,617306563,1256452642,2555937826

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

%C 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]

%F 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]

%e 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]

%t 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 *)

%Y Cf. A101509

%K nonn

%O 1,2

%A _Lior Manor_, Apr 09 2009

%E More terms from _Joel B. Lewis_, May 07 2009