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

0,13

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).

Links

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

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.

Example

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, ...

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);

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).

Sequence in context: A140736 A284993 A292068 * A350942 A140056 A240239

Adjacent sequences: A378063 A378064 A378065 * A378067 A378068 A378069

Keyword

sign,tabl

Author

Peter Luschny, Nov 15 2024

Status

approved