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A330662
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Triangle read by rows: T(n,k) is the number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing.
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3
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0, 0, 1, 1, 0, 2, 16, 24, 12, 8, 744, 960, 576, 192, 48, 56256, 69120, 39360, 13440, 2880, 384, 6385920, 7580160, 4204800, 1420800, 316800, 46080, 3840, 1018114560, 1178956800, 642539520, 216115200, 49190400, 7741440, 806400, 46080
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OFFSET
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0,6
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COMMENTS
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Rotations and reflections are counted separately.
By "2*n-sided polygons" we mean the polygons that can be drawn by connecting 2*n equally spaced points on a circle.
T(0,0)=0 and T(0,1)=1 by convention.
The sequence is limited to even-sided polygons, since all odd-sided polygons have no side passing through the center.
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LINKS
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FORMULA
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T(n,n) = 2^(n-1) * (n-1)! for all n >= 1.
T(n,k) = binomial(n,k)* Sum_{i=k..n} (-1)^(i-k)*binomial(n-k,i-k)*(2n-1-i)!*2^(i-1), for n>=2 and 0<=k<=n.
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EXAMPLE
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Triangle begins:
0;
0, 1;
1, 0, 2;
16, 24, 12, 8;
744, 960, 576, 192, 48;
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MAPLE
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T := (n, k) -> `if`(n<2, k, 2^(k-1)*binomial(n, k)*(2*n-k-1)!*hypergeom([k-n], [k-2*n+ 1], -2)):
seq(seq(simplify(T(n, k)), k=0..n), n=0..7); # Peter Luschny, Jan 07 2020
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CROSSREFS
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Row sums give A001710(2*n-1) (number of polygons with 2*n sides).
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KEYWORD
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AUTHOR
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STATUS
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approved
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