A371688 Triangle read by rows: T(n, k) = (2*n + 1)! * [y^(2*k)] [x^(2*n+1)] arctan(sec(x*y)*sinh(x)).

1, -1, 3, 5, -50, 25, -61, 1281, -2135, 427, 1385, -49860, 174510, -116340, 12465, -50521, 2778655, -16671930, 23340702, -8335965, 555731, 2702765, -210815670, 1932476975, -4637944740, 3478458555, -772990790, 35135945

Offset: 0

Author

Peter Luschny, Apr 03 2024

Keywords

• sign
• tabl

Comments

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

Examples

			Triangle starts:
  [0]      1;
  [1]     -1,       3;
  [2]      5,     -50,        25;
  [3]    -61,    1281,     -2135,      427;
  [4]   1385,  -49860,    174510,  -116340,    12465;
  [5] -50521, 2778655, -16671930, 23340702, -8335965, 555731;
		

Crossrefs

Cf. A000364 (column 0), A009843 (main diagonal), A012816 (row sums), A002436 (alternating row sums).

Programs

Maple

egf := arctan(sec(x*y)*sinh(x)):
serx := simplify(series(egf, x, 26)): coeffx := n -> n!*coeff(serx, x, n):
seq(lprint(seq(coeff(coeffx(2*n + 1), y, 2*k), k = 0..n)), n = 0..7);

Mathematica

T[n_,k_]:=(-1)^k*Binomial[2*n+1,2*k]*EulerE[2*n];Flatten[Table[T[n,k],{n,0,6},{k,0,n}]] (* Detlef Meya, Apr 07 2024 *)

Formula

T(n, k) = (-1)^k*binomial(2*n + 1, 2*k)*Euler(2*n). - Detlef Meya, Apr 07 2024