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, 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

Table of n, a(n) for n=0..54.

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

Adjacent sequences: A353950 A353951 A353952 * A353954 A353955 A353956

Keyword

nonn,tabl

Author

Michel Marcus, May 12 2022

Status

approved