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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. 2
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, 153, 495, 100, 66 (list; table; graph; refs; listen; history; text; internal format)
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.

G.f. for the n=6 row: (-10*x^7 + 6*x^6 + 6*x^5 + 6*x^4 + 5*x^3 + 8*x^2 + 4*x + 1)/(x - 1)^2.

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, 153, 179, 205, 231, ...

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), 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

STATUS

approved

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Last modified September 21 20:10 EDT 2021. Contains 347598 sequences. (Running on oeis4.)