A058001 Number of 3 X 3 matrices with entries mod n, up to row and column permutation.

1, 36, 738, 8240, 57675, 289716, 1144836, 3780288, 10865205, 27969700, 65834406, 143887536, 295467263, 575308020, 1069960200, 1911933696, 3298486761, 5516122788, 8972008810, 14233690800, 22078652211, 33555443636, 50058302988, 73417387200, 106006948125

Offset

1,2

Comments

Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Marko R. Riedel, Number of equivalence classes of matrices, Math Stackexchange.

Marko R. Riedel, Computing the cycle index for arbitary k x l matrices using Maple

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Formula

a(n) = (1/3!^2)*(n^9 + 6*n^6 + 9*n^5 + 8*n^3 + 12*n^2).

G.f.: x*(12*x^7+369*x^6+2514*x^5+4375*x^4+2360*x^3+423*x^2+26*x+1) / (x-1)^10. - Colin Barker, Jul 09 2013

Mathematica

CoefficientList[Series[x (12x^7+369x^6+2514x^5+4375x^4+2360x^3+423x^2+26x+1)/(x-1)^10, {x, 0, 30}], x] (* or *) LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 36, 738, 8240, 57675, 289716, 1144836, 3780288, 10865205}, 30] (* Harvey P. Dale, Nov 23 2024 *)

Crossrefs

Cf. A058002, A058003, A058004, A002724, A052271, A052272.

Row n=3 of A246106.

Sequence in context: A223998 A109405 A064541 * A004329 A089909 A049434

Adjacent sequences: A057998 A057999 A058000 * A058002 A058003 A058004

Keyword

easy,nonn

Author

Vladeta Jovovic, Nov 04 2000

Status

approved