A316623 Array read by antidiagonals: T(n,k) is the order of the group SL(n,Z_k).

1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 24, 168, 1, 1, 1, 48, 5616, 20160, 1, 1, 1, 120, 43008, 12130560, 9999360, 1, 1, 1, 144, 372000, 660602880, 237783237120, 20158709760, 1, 1, 1, 336, 943488, 29016000000, 167761422581760, 42064805779476480, 163849992929280, 1

Offset

0,9

Comments

All rows are multiplicative.

Equivalently, the number of n X n matrices mod k with determinant 1.

Also, for k prime (but not higher prime powers) the number of n X n matrices over GF(k) with determinant 1.

Links

Table of n, a(n) for n=0..44.

R. P. Brent and B. D. McKay, Determinants and ranks of random matrices over Zm, Discrete Mathematics 66 (1987) pp. 35-49.

J. M. Lockhart and W. P. Wardlaw, Determinants of Matrices over the Integers Modulo m, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 207-214.

The Group Properties Wiki, Order formulas for linear groups

Formula

T(n,p^e) = (p^e)^(n^2-1) * Product_{j=2..n} (1 - 1/p^j) for prime p, n > 0.

Example

Array begins:
==============================================================
n\k| 1       2        3         4           5           6
---+----------------------------------------------------------
0  | 1       1        1         1           1            1 ...
1  | 1       1        1         1           1            1 ...
2  | 1       6       24        48         120          144 ...
3  | 1     168     5616     43008      372000       943488 ...
4  | 1   20160 12130560 660602880 29016000000 244552089600 ...
5  | 1 9999360 ...
...

Mathematica

T[n_, k_] := If[k == 1 || n == 0, 1, k^(n^2-1) Product[1 - p^-j, {p, FactorInteger[k][[All, 1]]}, {j, 2, n}]];
Table[T[n-k+1, k], {n, 0, 8}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Sep 19 2019 *)

Prog

(GAP)
T:=function(n, k) if k=1 or n=0 then return 1; else return Order(SL(n, Integers mod k)); fi; end;
for n in [0..5] do Print(List([1..6], k->T(n, k)), "\n"); od;
(PARI) T(n, k)={my(f=factor(k)); if(n<1, n==0, k^(n^2-1) * prod(i=1, #f~, my(p=f[i, 1]); prod(j=2, n, (1 - p^(-j)))))}

Crossrefs

Rows n=2..4 are A000056, A011785, A011786.

Columns k=2..5, 7 are A002884, A003787, A011787, A003789, A003790.

Cf. A316622.

Sequence in context: A290480 A183092 A050449 * A108131 A073354 A358461

Adjacent sequences: A316620 A316621 A316622 * A316624 A316625 A316626

Keyword

nonn,mult,tabl

Author

Andrew Howroyd, Jul 08 2018

Status

approved