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

1,5

Links

Max Alekseyev, Table of n, a(n) for n = 1..351

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)

Example

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, ...
  ...

Crossrefs

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).

Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).

Diagonal: A299754 (n=k).

Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.

Sequence in context: A180171 A140822 A212954 * A089239 A223968 A214846

Adjacent sequences: A299804 A299805 A299806 * A299808 A299809 A299810

Keyword

nonn,tabl

Author

Max Alekseyev, Feb 24 2018

Extensions

Row 6 corrected by Max Alekseyev, Aug 14 2022

Status

approved