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

%I #9 Apr 07 2024 03:36:37

%S 1,-1,3,5,-50,25,-61,1281,-2135,427,1385,-49860,174510,-116340,12465,

%T -50521,2778655,-16671930,23340702,-8335965,555731,2702765,-210815670,

%U 1932476975,-4637944740,3478458555,-772990790,35135945

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

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

%F T(n, k) = (-1)^k*binomial(2*n + 1, 2*k)*Euler(2*n). - _Detlef Meya_, Apr 07 2024

%e Triangle starts:

%e [0] 1;

%e [1] -1, 3;

%e [2] 5, -50, 25;

%e [3] -61, 1281, -2135, 427;

%e [4] 1385, -49860, 174510, -116340, 12465;

%e [5] -50521, 2778655, -16671930, 23340702, -8335965, 555731;

%p egf := arctan(sec(x*y)*sinh(x)):

%p serx := simplify(series(egf, x, 26)): coeffx := n -> n!*coeff(serx, x, n):

%p seq(lprint(seq(coeff(coeffx(2*n + 1), y, 2*k), k = 0..n)), n = 0..7);

%t 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 *)

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

%K sign,tabl

%O 0,3

%A _Peter Luschny_, Apr 03 2024