A378066 Array read by ascending antidiagonals: A(n, k) = (-2*n)^k * Euler(k, (n - 1)/(2*n)) for n >= 1 and A(0, k) = 1.

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -3, -2, 1, 1, 1, -8, -11, 0, 1, 1, 1, -15, -26, 57, 16, 1, 1, 1, -24, -47, 352, 361, 0, 1, 1, 1, -35, -74, 1185, 1936, -2763, -272, 1, 1, 1, -48, -107, 2976, 6241, -38528, -24611, 0, 1

Offset: 0

Author

Peter Luschny, Nov 15 2024

Keywords

• sign
• tabl

Comments

This is the counterpart of A377666, where A(1, n) are the secant numbers A122045(n). Here A(1, n) are the tangent numbers A155585(n).

Examples

			Array starts:
  [0]  1, 1,   1,    1,     1,     1,        1, ...  A000012
  [1]  1, 1,   0,   -2,     0,    16,        0, ...  A155585
  [2]  1, 1,  -3,  -11,    57,   361,    -2763, ...  A188458
  [3]  1, 1,  -8,  -26,   352,  1936,   -38528, ...  A000810
  [4]  1, 1, -15,  -47,  1185,  6241,  -230895, ...  A000813
  [5]  1, 1, -24,  -74,  2976, 15376,  -906624, ...  A378065
  [6]  1, 1, -35, -107,  6265, 32041, -2749355, ...
  [7]  1, 1, -48, -146, 11712, 59536, -6997248, ...
		

Crossrefs

Rows: A000012, A155585, A188458, A000810, A000813, A378065.

Columns: A005563 (k=2), A080663 (k=3), A378064 (k=4).

Cf. A378063 (main diagonal), A377666 (secant), A081658 (column generating polynomials).

Programs

Maple

A := (n, k) -> ifelse(n = 0, 1, (-2*n)^k * euler(k, (n - 1) / (2*n))):
for n from 0 to 7 do seq(A(n, k), k = 0..9) od; # row by row
# Alternative:
A := proc(n, k) local j; add(binomial(k, j)*euler(j, 1/2)*(-2*n)^j, j = 0..k) end: seq(seq(A(n - k, k), k = 0..n), n = 0..10);
# Using generating functions:
egf := n -> exp(x)/cosh(n*x): ser := n -> series(egf(n), x, 14):
row := n -> local k; seq(k!*coeff(ser(n), x, k), k = 0..7):
seq(lprint(row(n)), n = 0..7);

Formula

A(n, k) = k! * [x^k] exp(x)/cosh(n*x).

A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 1/2) *(-2*n)^j.