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A052365 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, under row and column permutations.

%I A052365 #18 Sep 03 2019 09:56:30
%S A052365 1,1,4,10,24,51,114,219,424,768,1352,2278,3759,5978,9328,14181,21164,
%T A052365 30943,44560,63063,88088,121321,165152,222157,295857,389948,509456,
%U A052365 659697,847552,1080452,1367814,1719652,2148596,2668107,3294676,4046069
%N A052365 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, under row and column permutations.
%C A052365 Also Molien series for group of structure S_3 X S_3 = (Z_3 X Z_3).O_2^+(3) and order 36, corresponding to complete weight enumerators of Hermitian self-dual GF(3)-linear codes over GF(9) containing the all-ones vector.
%H A052365 Andrew Howroyd, <a href="/A052365/b052365.txt">Table of n, a(n) for n = 0..1000</a>
%H A052365 G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H A052365 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%F A052365 G.f.: -(x^10+2*x^8+x^7+7*x^6-3*x^5+4*x^3+x^2-2*x+1) / ((x^4-x^3+x-1)*(x^3-1)^3*(x+1)^3*(x-1)^5).
%F A052365 Another form for g.f.: u1/u2, where u1 := 1 + x + 2*x^3 + 10*x^4 + 17*x^5 + 19*x^6 + 20*x^7 + 29*x^8 + 37*x^9 + 34*x^10 + 23*x^11 + 12*x^12 + 7*x^13 + 3*x^14 + x^15 u2 := (1-x^2)^4*(1-x^3)^4*(1-x^6);
%t A052365 permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A052365 c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
%t A052365 M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
%t A052365 a[n_] := M[3, 3, n];
%t A052365 a /@ Range[0, 40] (* _Jean-François Alcover_, Sep 03 2019, after _Andrew Howroyd_ in A318795 *)
%Y A052365 Row 3 of A318795.
%Y A052365 Cf. A002724, A053307, A052366, A052267, A092091.
%K A052365 nonn
%O A052365 0,3
%A A052365 _Vladeta Jovovic_, Mar 08 2000