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A377067 Number of n X 3 0..2 matrices with row sums 3 and column sums n up to permutations of rows.

%I A377067 #8 Oct 15 2024 15:46:55
%S A377067 1,1,4,6,12,18,30,42,63,85,118,154,204,258,330,408,507,615,748,892,
%T A377067 1066,1254,1476,1716,1995,2295,2640,3010,3430,3880,4386,4926,5529,
%U A377067 6171,6882,7638,8470,9352,10318,11340,12453,13629,14904,16248,17700,19228,20872,22600,24453,26397,28476
%N A377067 Number of n X 3 0..2 matrices with row sums 3 and column sums n up to permutations of rows.
%C A377067 Also, the number of n X 3 {-1,0,1} matrices with all rows and columns summing to zero up to permutations of rows.
%H A377067 Andrew Howroyd, <a href="/A377067/b377067.txt">Table of n, a(n) for n = 0..1000</a>
%H A377067 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,-1,1,3,-1,-2,1).
%F A377067 G.f.: (2/(1 - x^3) - 1)/((1 - x)*(1 - x^2)^3).
%F A377067 G.f.: (1 - x + x^2)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)).
%e A377067 The a(2) = 4 matrices are:
%e A377067   [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]
%e A377067   [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]
%e A377067 The a(3) = 6 matrices are:
%e A377067   [1 1 1]  [2 1 0]  [2 0 1]  [1 2 0]  [2 1 0]  [2 0 1]
%e A377067   [1 1 1]  [0 1 2]  [0 2 0]  [1 0 2]  [1 0 2]  [1 2 0]
%e A377067   [1 1 1]  [1 1 1]  [1 1 1]  [1 1 1]  [0 2 1]  [0 1 2]
%o A377067 (PARI) Vec((1 - x + x^2)/((1 - x)^5*(1 + x)^2*(1 + x + x^2)) + O(x^51))
%Y A377067 Column k=3 of A377063.
%Y A377067 Cf. A172634, A377065, A377068.
%K A377067 nonn,easy
%O A377067 0,3
%A A377067 _Andrew Howroyd_, Oct 15 2024