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%I #34 Aug 15 2022 15:31:06
%S 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,
%T 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,
%U 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
%N 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.
%H Max Alekseyev, <a href="/A299807/b299807.txt">Table of n, a(n) for n = 1..351</a>
%F From _Chai Wah Wu_, May 28 2018: (Start)
%F The following are all conjectures.
%F 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).
%F 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.
%F (End)
%e Array starts:
%e n=1: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e n=2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
%e n=3: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
%e n=4: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...
%e n=5: 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, ...
%e n=6: 1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, ...
%e n=7: 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, ...
%e n=8: 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, ...
%e n=9: 1, 9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ...
%e n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951, 6531, 10201, 15231, ...
%e ...
%Y Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8).
%Y Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).
%Y Diagonal: A299754 (n=k).
%Y Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.
%K nonn,tabl
%O 1,5
%A _Max Alekseyev_, Feb 24 2018
%E Row 6 corrected by _Max Alekseyev_, Aug 14 2022
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