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A212801
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Square array read by antidiagonals: T(m,n) = number of Eulerian circuits in the Cartesian product of two directed cycles of lengths m and n.
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7
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1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 108, 40, 1, 1, 121, 793, 793, 121, 1, 1, 364, 5611, 12800, 5611, 364, 1, 1, 1093, 39312, 193721, 193721, 39312, 1093, 1, 1, 3280, 274933, 2886520, 6050000, 2886520, 274933, 3280, 1, 1, 9841, 1923025, 42999713, 183990301, 183990301, 42999713, 1923025, 9841, 1
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OFFSET
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1,5
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COMMENTS
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All rows and columns are given by linear recurrences with constant coefficients. Empirically, the order of the recurrences for n=1..8 appear to be 1, 2, 4, 8, 16, 24, 64, 128. - Andrew Howroyd, May 19 2020
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LINKS
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FORMULA
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T(m,n) = Product_{k=1..n-1} Product_{h=1..m-1} (2 - exp(2*i*h*Pi/m) - exp(2*i*k*Pi/n)), where i is the imaginary unit.
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EXAMPLE
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Array begins:
======================================================================
m\n| 1 2 3 4 5 6 7
---|------------------------------------------------------------------
1 | 1 1 1 1 1 1 1 ...
2 | 1 4 13 40 121 364 1093 ...
3 | 1 13 108 793 5611 39312 274933 ...
4 | 1 40 793 12800 193721 2886520 42999713 ...
5 | 1 121 5611 193721 6050000 183990301 5598183221 ...
6 | 1 364 39312 2886520 183990301 11218701312 681838513861 ...
7 | 1 1093 274933 42999713 5598183221 681838513861 81959473720768 ...
...
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MATHEMATICA
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T[m_, n_] := Product[2 - Exp[2*I*h*Pi/m] - Exp[2*I*k*Pi/n], {h, 1, m - 1}, {k, 1, n - 1}];
Table[T[m - n + 1, n] // FullSimplify, {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 30 2018 *)
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PROG
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(PARI) T(m, n) = prod(k=1, n-1, prod(h=1, m-1, 2 - exp(2*I*h*Pi/m) - exp(2*I*k*Pi/n)));
tabl(nn) = matrix(nn, nn, m, n, round(real(T(m, n)))); \\ Michel Marcus, Feb 01 2016
(PARI) \\ all integer version.
R(n, f)={my(p=lift(prod(i=1, n-1, f(Mod(x^i, 1-x^n))))); sumdiv(n, d, moebius(n/d) * polcoef(p, d%n, x))}
T(m, n)={my(p=R(n, x->2-x-y)); R(m, x->subst(p, y, x))} \\ Andrew Howroyd, May 19 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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