cp's OEIS Frontend

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

%I A316623 #11 Sep 19 2019 14:38:10
%S A316623 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,
%T A316623 12130560,9999360,1,1,1,144,372000,660602880,237783237120,20158709760,
%U A316623 1,1,1,336,943488,29016000000,167761422581760,42064805779476480,163849992929280,1
%N A316623 Array read by antidiagonals: T(n,k) is the order of the group SL(n,Z_k).
%C A316623 All rows are multiplicative.
%C A316623 Equivalently, the number of n X n matrices mod k with determinant 1.
%C A316623 Also, for k prime (but not higher prime powers) the number of n X n matrices over GF(k) with determinant 1.
%H A316623 R. P. Brent and B. D. McKay, <a href="https://doi.org/10.1016/0012-365X(87)90117-8">Determinants and ranks of random matrices over Zm</a>, Discrete Mathematics 66 (1987) pp. 35-49.
%H A316623 J. M. Lockhart and W. P. Wardlaw, <a href="https://www.jstor.org/stable/27643029">Determinants of Matrices over the Integers Modulo m</a>, Mathematics Magazine, Vol. 80, No. 3 (Jun., 2007), pp. 207-214.
%H A316623 The Group Properties Wiki, <a href="https://groupprops.subwiki.org/wiki/Order_formulas_for_linear_groups">Order formulas for linear groups</a>
%F A316623 T(n,p^e) = (p^e)^(n^2-1) * Product_{j=2..n} (1 - 1/p^j) for prime p, n > 0.
%e A316623 Array begins:
%e A316623 ==============================================================
%e A316623 n\k| 1       2        3         4           5           6
%e A316623 ---+----------------------------------------------------------
%e A316623 0  | 1       1        1         1           1            1 ...
%e A316623 1  | 1       1        1         1           1            1 ...
%e A316623 2  | 1       6       24        48         120          144 ...
%e A316623 3  | 1     168     5616     43008      372000       943488 ...
%e A316623 4  | 1   20160 12130560 660602880 29016000000 244552089600 ...
%e A316623 5  | 1 9999360 ...
%e A316623 ...
%t A316623 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}]];
%t A316623 Table[T[n-k+1, k], {n, 0, 8}, {k, n+1, 1, -1}] // Flatten (* _Jean-François Alcover_, Sep 19 2019 *)
%o A316623 (GAP)
%o A316623 T:=function(n,k) if k=1 or n=0 then return 1; else return Order(SL(n, Integers mod k)); fi; end;
%o A316623 for n in [0..5] do Print(List([1..6], k->T(n,k)), "\n"); od;
%o A316623 (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)))))}
%Y A316623 Rows n=2..4 are A000056, A011785, A011786.
%Y A316623 Columns k=2..5, 7 are A002884, A003787, A011787, A003789, A003790.
%Y A316623 Cf. A316622.
%K A316623 nonn,mult,tabl
%O A316623 0,9
%A A316623 _Andrew Howroyd_, Jul 08 2018