A353952 Array T(n,k) = D(2*n, -2*k-1), where D(i,j) are the polysecant numbers, for n,k >= 0, read by antidiagonals.

1, 1, 1, 1, 13, 1, 1, 121, 121, 1, 1, 1093, 4081, 1093, 1, 1, 9841, 111721, 111721, 9841, 1, 1, 88573, 2880481, 7256173, 2880481, 88573, 1, 1, 797161, 72799321, 403087441, 403087441, 72799321, 797161, 1, 1, 7174453, 1827068881, 20966597653, 42931692481, 20966597653, 1827068881, 7174453, 1

Offset

0,5

Links

Table of n, a(n) for n=0..44.

Masanobu Kaneko, Maneka Pallewatta, and Hirofumi Tsumura, On Polycosecant Numbers, J. Integer Seq. 23 (2020), no. 6, 17 pp. See Table 2 p. 9.

Kyosuke Nishibiro, On some properties of polycosecant numbers and polycotangent numbers, arXiv:2205.05247 [math.NT], 2022.

Formula

D(n, k) = Sum_{m=0..floor(n/2)} (1/(2*m+1)^(k+1))*Sum_{p=2*m..n} (-1)^p*(p+1)!*binomial(p, 2*m)*Stirling2(n+1,p+1)/2^p)).

Example

The array begins:
  1    1       1         1           1 ...
  1   13     121      1093        9841 ...
  1  121    4081    111721     2880481 ...
  1 1093  111721   7256173   403087441 ...
  1 9841 2880481 403087441 42931692481 ...
  ...

Prog

(PARI) D(n, k) = sum(m=0, n\2, (1/(2*m+1)^(k+1))*sum(p=2*m, n, (-1)^p*(p+1)!*binomial(p, 2*m)*stirling(n+1, p+1, 2)/2^p));
matrix(5, 5, n, k, n--; k--; D(2*n, -2*k-1))

Crossrefs

Cf. A008277 (Stirling2), A001896, A001897, A353953.

Sequence in context: A174731 A342891 A174694 * A340432 A156539 A172300

Adjacent sequences: A353949 A353950 A353951 * A353953 A353954 A353955

Keyword

nonn,tabl

Author

Michel Marcus, May 12 2022

Status

approved