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A316622
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Array read by antidiagonals: T(n,k) is the order of the group GL(n,Z_k).
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11
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1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 2, 48, 168, 1, 1, 4, 96, 11232, 20160, 1, 1, 2, 480, 86016, 24261120, 9999360, 1, 1, 6, 288, 1488000, 1321205760, 475566474240, 20158709760, 1, 1, 4, 2016, 1886976, 116064000000, 335522845163520, 84129611558952960, 163849992929280, 1
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OFFSET
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0,8
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COMMENTS
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All rows are multiplicative.
Equivalently, the number of invertible n X n matrices mod k.
Also, for k prime (but not higher prime powers) the number of nonsingular n X n matrices over GF(k).
For k >= 2, n! divides T(n,k) since the subgroup of GL(n,k) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). Note that a permutation matrix is an orthogonal matrix, hence having determinant +-1. - Jianing Song, Oct 29 2022
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LINKS
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FORMULA
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T(n,p^e) = (p^e)^(n^2) * Product_{j=1..n} (1 - 1/p^j) for prime p.
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EXAMPLE
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Array begins:
=================================================================
n\k| 1 2 3 4 5 6
---+-------------------------------------------------------------
0 | 1 1 1 1 1 1 ...
1 | 1 1 2 2 4 2 ...
2 | 1 6 48 96 480 288 ...
3 | 1 168 11232 86016 1488000 1886976 ...
4 | 1 20160 24261120 1321205760 116064000000 489104179200 ...
5 | 1 9999360 ...
...
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MATHEMATICA
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T[_, 1] = T[0, _] = 1; T[n_, k_] := T[n, k] = Module[{f = FactorInteger[k], p, e}, If[Length[f] == 1, {p, e} = f[[1]]; (p^e)^(n^2)* Product[(1 - 1/p^j), {j, 1, n}], Times @@ (T[n, Power @@ #]& /@ f)]];
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PROG
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(GAP)
T:=function(n, k) if k=1 or n=0 then return 1; else return Order(GL(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)); k^(n^2) * prod(i=1, #f~, my(p=f[i, 1]); prod(j=1, n, (1 - p^(-j))))}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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