A302453 a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. tan(x).

1, 4, 168, 26400, 10373760, 8226518272, 11524607732736, 26047611675267072, 88935060882222120960, 436394080487109570265088, 2959343413232671759344861184, 26874522377891724867898947141632, 318464577992023576681854032513335296, 4818779071094868918454887699722367139840

Offset: 1

Author

Ilya Gutkovskiy, Apr 08 2018

Keywords

• nonn

Comments

a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. tanh(x) (absolute values).

Examples

			The initial coefficients of successive iterations of e.g.f. A(x) = tan(x) (odd powers only) are as follows:
n = 1: (1),  2,   16,    272,      7936,  ... e.g.f. A(x)
n = 2:  1,  (4),  72,   2896,    203904,  ... e.g.f. A(A(x))
n = 3:  1,   6, (168), 10672,   1198080,  ... e.g.f. A(A(A(x)))
n = 4:  1,   8,  304, (26400),  4071040,  ... e.g.f. A(A(A(A(x))))
n = 5:  1,  10,  480,  52880, (10373760), ... e.g.f. A(A(A(A(A(x)))))
...
More explicitly, the successive iterations of e.g.f. A(x) = tan(x) begin:
tan(x) = x/1! + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + 7936*x^9/9! + ...
tan(tan(x)) = x/1! + 4*x^3/3! + 72*x^5/5! + 2896*x^7/7! + 203904*x^9/9! + ...
tan(tan(tan(x))) = x/1! + 6*x^3/3! + 168*x^5/5! + 10672*x^7/7! + 1198080*x^9/9! + ...
tan(tan(tan(tan(x)))) = x/1! + 8*x^3/3! + 304*x^5/5! + 26400*x^7/7! + 4071040*x^9/9! + ...
tan(tan(tan(tan(tan(x))))) = x/1! + 10*x^3/3! + 480*x^5/5! + 52880*x^7/7! + 10373760*x^9/9! + ...
		

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000182, A003718, A003720.

Programs

Mathematica

Table[(2 n - 1)! SeriesCoefficient[Nest[Function[x, Tan[x]], x, n], {x, 0, 2 n - 1}], {n, 14}]