A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 10, 5, 1, 1, 6, 15, 16, 15, 6, 1, 1, 7, 19, 35, 25, 21, 7, 1, 1, 8, 28, 37, 70, 36, 28, 8, 1, 1, 9, 33, 84, 61, 126, 49, 36, 9, 1, 1, 10, 45, 96, 210, 91, 210, 64, 45, 10, 1, 1, 11, 51, 163, 225, 462, 127, 330, 81, 55, 11, 1, 1, 12, 66, 180, 477, 456, 924, 169, 495, 100, 66
Offset: 1
Examples
Array starts: n=1: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... n=2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... n=3: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ... n=4: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ... n=5: 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, ... n=6: 1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, ... n=7: 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, ... n=8: 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, ... n=9: 1, 9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ... n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951, 6531, 10201, 15231, ... ...
Links
- Max Alekseyev, Table of n, a(n) for n = 1..351
Crossrefs
Formula
From Chai Wah Wu, May 28 2018: (Start)
The following are all conjectures.
For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1).
For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p.
(End)
Extensions
Row 6 corrected by Max Alekseyev, Aug 14 2022