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A353953
Array T(n,k) = beta(2*n, -k), where beta(i,j) are the polycotangent numbers, for n,k >= 0, read by ascending antidiagonals.
1
1, 1, 1, 1, 2, 1, 1, 8, 5, 1, 1, 32, 41, 14, 1, 1, 128, 365, 200, 41, 1, 1, 512, 3281, 3104, 977, 122, 1, 1, 2048, 29525, 49280, 23801, 4808, 365, 1, 1, 8192, 265721, 786944, 589217, 174752, 23801, 1094, 1, 1, 32768, 2391485, 12584960, 14677961, 6297728, 1257125, 118280, 3281, 1
OFFSET
0,5
LINKS
Masanobu Kaneko, Maneka Pallewatta, and Hirofumi Tsumura, On Polycosecant Numbers, J. Integer Seq. 23 (2020), no. 6, 17 pp.
Kyosuke Nishibiro, On some properties of polycosecant numbers and polycotangent numbers, arXiv:2205.05247 [math.NT], 2022.
EXAMPLE
The array begins:
1 1 1 1 1 ...
1 2 5 14 41 ...
1 8 41 200 977 ...
1 32 365 3104 23801 ...
1 128 3281 49280 589217 ...
PROG
(PARI) beta(n, k) = if (!(n%2), n>>=1; sum(j=0, 2*n, sum(i=0, j\2, (-1)^j*j!*binomial(j+1, 2*i+1)*((j+1)*(j+2)*stirling(2*n, j+2, 2)/2+stirling(2*n+1, j+1, 2))/(2^j*(2*i+1)^k))));
matrix(5, 5, n, k, n--; k--; beta(2*n, -k))
CROSSREFS
Sequence in context: A297733 A255812 A249141 * A102875 A329070 A157785
KEYWORD
nonn,tabl
AUTHOR
Michel Marcus, May 12 2022
STATUS
approved