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%I #23 Sep 07 2018 11:57:35
%S 1,3,14,44,129,316,714,1452,2775,4963,8478,13838,21827,33306,49504,
%T 71754,101871,141807,194128,261570,347633,456026,591384,758596,963657,
%U 1212861,1513806,1874440,2304225,2813030,3412466,4114608,4933519
%N Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.
%H Andrew Howroyd, <a href="/A058389/b058389.txt">Table of n, a(n) for n = 0..1000</a>
%H V. Jovovic, <a href="/A058389/a058389.pdf">Illustration of initial terms</a>
%H V. Jovovic, <a href="/A058389/a058389a.pdf">Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.</a>
%F a(n) = (1/6)*(C(C(n + 2, 2) + 2, 3) + 3/2*floor((n + 2)/2)*(C(n + 2, 2) - floor((n + 2)/2)) + 3*C(floor((n + 2)/2) + 2, 3) + 2*floor(C(n + 2, 2)/3) + 2*C(C(n + 2, 2) - 3*floor(C(n + 2, 2)/3) + 2, 3)).
%F Empirical G.f.: -(x^8 + 3*x^7 + 14*x^6 + 12*x^5 + 15*x^4 + 9*x^3 + 5*x^2 + 1) / ((x-1)^7*(x+1)^3*(x^2+x+1)). - _Colin Barker_, Dec 27 2012
%t a[n_] := (m = Mod[n, 6]; (n^3 + 9*n^2 + 39*n + 120)*n^3 + Which[m == 0, 12*(23*n^2 + 32*n + 24), m == 1 || m == 5, 249*n^2 + 303*n + 143, m == 2 || m == 4, 4*(69*n^2 + 96*n + 56), m == 3, 3*(83*n^2 + 101*n + 69)])/288; Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Oct 12 2011, after _Vladeta Jovovic_ *)
%o (PARI) \\ See A318951 for RowSumMats
%o a(n)=RowSumMats(3, 3, n); \\ _Andrew Howroyd_, Sep 05 2018
%Y Row 3 of A318951.
%Y Cf. A002817, A052282, A058390, A058391, A058392.
%Y Cf. A001501, A050535, A050913, A058528, A058783, A058784, A058785.
%K nice,nonn,easy
%O 0,2
%A _Vladeta Jovovic_, Nov 24 2000
%E More terms from _Marc LeBrun_, Dec 11 2000
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