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

1, 1, 4, 10, 24, 51, 114, 219, 424, 768, 1352, 2278, 3759, 5978, 9328, 14181, 21164, 30943, 44560, 63063, 88088, 121321, 165152, 222157, 295857, 389948, 509456, 659697, 847552, 1080452, 1367814, 1719652, 2148596, 2668107, 3294676, 4046069

Offset

0,3

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Index entries for Molien series

Formula

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).

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);

Mathematica

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];
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}];
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!)];
a[n_] := M[3, 3, n];
a /@ Range[0, 40] (* Jean-François Alcover, Sep 03 2019, after Andrew Howroyd in A318795 *)

Crossrefs

Row 3 of A318795.

Cf. A002724, A053307, A052366, A052267, A092091.

Sequence in context: A022569 A093831 A274582 * A107659 A162588 A280541

Adjacent sequences: A052362 A052363 A052364 * A052366 A052367 A052368

Keyword

nonn

Author

Vladeta Jovovic, Mar 08 2000

Status

approved