A371687 Triangle read by rows: T(n, k) = (-1)^(n-k) * (2*n + 1)! * [y^(2*k)] [x^(2*n+1)] arctan(sec(x*y)*tanh(x)).

1, 4, 3, 80, 80, 25, 3904, 5376, 2660, 427, 354560, 626688, 433440, 131712, 12465, 51733504, 111738880, 99242880, 43804992, 9021540, 555731, 11070525440, 28258074624, 30647302400, 17666508288, 5509286640, 816337808, 35135945

Offset

0,2

Comments

Expansion of the exponential generating function arctan(sec(x*y)*tanh(x)), nonzero terms only.

Links

Table of n, a(n) for n=0..27.

Example

Triangle starts:
  [0]        1;
  [1]        4,         3;
  [2]       80,        80,       25;
  [3]     3904,      5376,     2660,      427;
  [4]   354560,    626688,   433440,   131712,   12465;
  [5] 51733504, 111738880, 99242880, 43804992, 9021540, 555731;

Maple

egf := arctan(sec(x*y)*tanh(x)):
serx := simplify(series(egf, x, 26)): coeffx := n -> n!*coeff(serx, x, n):
seq(print(seq((-1)^(n-k)*coeff(coeffx(2*n+1), y, 2*k), k = 0..n)), n = 0..6);

Crossrefs

Cf. A002436 (column 0), A009843 (main diagonal), A012798 (row sums), A012835 (alternating row sums).

Cf. A371688.

Sequence in context: A298314 A299389 A220556 * A299188 A300026 A349589

Adjacent sequences: A371684 A371685 A371686 * A371688 A371689 A371690

Keyword

nonn,tabl

Author

Peter Luschny, Apr 03 2024

Status

approved