A371687 - OEIS

%I #5 Apr 04 2024 10:24:40

%S 1,4,3,80,80,25,3904,5376,2660,427,354560,626688,433440,131712,12465,

%T 51733504,111738880,99242880,43804992,9021540,555731,11070525440,

%U 28258074624,30647302400,17666508288,5509286640,816337808,35135945

%N 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)).

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

%e Triangle starts:

%e   [0]        1;

%e   [1]        4,         3;

%e   [2]       80,        80,       25;

%e   [3]     3904,      5376,     2660,      427;

%e   [4]   354560,    626688,   433440,   131712,   12465;

%e   [5] 51733504, 111738880, 99242880, 43804992, 9021540, 555731;

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

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

%p seq(print(seq((-1)^(n-k)*coeff(coeffx(2*n+1), y, 2*k), k = 0..n)), n = 0..6);

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

%Y Cf. A371688.

%K nonn,tabl

%O 0,2

%A _Peter Luschny_, Apr 03 2024