A330662 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.
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
Offset: 0
Examples
Triangle begins:
0;
0, 1;
1, 0, 2;
16, 24, 12, 8;
744, 960, 576, 192, 48;
Links
- Ludovic Schwob, Table of n, a(n) for n = 0..494
- Ludovic Schwob, Illustration of T(3,k), 0≤k≤3.
Crossrefs
Programs
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Maple
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
Formula
T(n,n) = 2^(n-1) * (n-1)! for all n >= 1.
T(n,0) = A307923(n) 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.
Comments