This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A159297 #13 Jul 22 2025 06:23:28
%S A159297 1,4,10,25,58,130,286,620,1329,2827,5977,12559,26227,54493,112849,
%T A159297 233272,481616,992955,2043238,4194649,8591014,17559133,35833948,
%U A159297 73054885,148849186,303171755,617306563,1256452642,2555937826
%N A159297 Number of 3D matrices with positive integer entries such that sum of all entries equals n.
%C A159297 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 A159297 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 A159297 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 A159297 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 A159297 Cf. A101509
%K A159297 nonn
%O A159297 1,2
%A A159297 _Lior Manor_, Apr 09 2009
%E A159297 More terms from _Joel B. Lewis_, May 07 2009