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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, -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
0,3
COMMENTS
Expansion of the exponential generating function arctan(sec(x*y)*sinh(x)), nonzero terms only.
FORMULA
T(n, k) = (-1)^k*binomial(2*n + 1, 2*k)*Euler(2*n). - Detlef Meya, Apr 07 2024
EXAMPLE
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;
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 *)
CROSSREFS
Cf. A000364 (column 0), A009843 (main diagonal), A012816 (row sums), A002436 (alternating row sums).
Sequence in context: A120426 A077201 A196467 * A219506 A171775 A260227
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Apr 03 2024
STATUS
approved