A302452 a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. sinh(x).

1, 2, 33, 2160, 368145, 130426016, 83303826249, 87104014381056, 139088689115885505, 321859857651846029824, 1036109938469605247521009, 4490275483028481600517832704, 25503692273369769781221175069521, 185636732310716855091866841134243840, 1699077450890747555020338066545506541145

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. sin(x) (absolute values).

Examples

			The initial coefficients of successive iterations of e.g.f. A(x) = sinh(x) (odd powers only) are as follows:
n = 1: (1), 1,    1,     1,       1,  ... e.g.f. A(x)
n = 2:  1, (2),  12,   128,    1872,  ... e.g.f. A(A(x))
n = 3:  1,  3,  (33),  731,   25857,  ... e.g.f. A(A(A(x)))
n = 4:  1,  4,   64, (2160)  121600,  ... e.g.f. A(A(A(A(x))))
n = 5:  1,  5,  105,  4765, (368145), ... e.g.f. A(A(A(A(A(x)))))
...
More explicitly, the successive iterations of e.g.f. A(x) = sinh(x) begin:
sinh(x) = x/1! + x^3/3! + x^5/5! + x^7/7! + x^9/9! + ...
sinh(sinh(x)) = x/1! + 2*x^3/3! + 12*x^5/5! + 128*x^7/7! + 1872*x^9/9! + ...
sinh(sinh(sinh(x))) = x/1! + 3*x^3/3! + 33*x^5/5! + 731*x^7/7! + 25857*x^9/9! + ...
sinh(sinh(sinh(sinh(x)))) = x/1! + 4*x^3/3! + 64*x^5/5! + 2160*x^7/7! + 121600*x^9/9! + ...
sinh(sinh(sinh(sinh(sinh(x))))) = x/1! + 5*x^3/3! + 105*x^5/5! + 4765*x^7/7! + 368145*x^9/9! + ...
		

Links

• N. J. A. Sloane, Transforms

Crossrefs

Cf. A003712, A003715.

Programs

Mathematica

Table[(2 n - 1)! SeriesCoefficient[Nest[Function[x, Sinh[x]], x, n], {x, 0, 2 n - 1}], {n, 15}]