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 A316623 Array read by antidiagonals: T(n,k) is the order of the group SL(n,Z_k). 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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: A183092 A050449 A260340 * A108131 A073354 A197420 Adjacent sequences:  A316620 A316621 A316622 * A316624 A316625 A316626 KEYWORD nonn,mult,tabl AUTHOR Andrew Howroyd, Jul 08 2018 STATUS approved

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Last modified October 18 00:51 EDT 2019. Contains 328135 sequences. (Running on oeis4.)