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

%I A302452 #4 Apr 09 2018 22:33:41
%S A302452 1,2,33,2160,368145,130426016,83303826249,87104014381056,
%T A302452 139088689115885505,321859857651846029824,1036109938469605247521009,
%U A302452 4490275483028481600517832704,25503692273369769781221175069521,185636732310716855091866841134243840,1699077450890747555020338066545506541145
%N A302452 a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. sinh(x).
%C A302452 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).
%H A302452 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%e A302452 The initial coefficients of successive iterations of e.g.f. A(x) = sinh(x) (odd powers only) are as follows:
%e A302452 n = 1: (1), 1,    1,     1,       1,  ... e.g.f. A(x)
%e A302452 n = 2:  1, (2),  12,   128,    1872,  ... e.g.f. A(A(x))
%e A302452 n = 3:  1,  3,  (33),  731,   25857,  ... e.g.f. A(A(A(x)))
%e A302452 n = 4:  1,  4,   64, (2160)  121600,  ... e.g.f. A(A(A(A(x))))
%e A302452 n = 5:  1,  5,  105,  4765, (368145), ... e.g.f. A(A(A(A(A(x)))))
%e A302452 ...
%e A302452 More explicitly, the successive iterations of e.g.f. A(x) = sinh(x) begin:
%e A302452 sinh(x) = x/1! + x^3/3! + x^5/5! + x^7/7! + x^9/9! + ...
%e A302452 sinh(sinh(x)) = x/1! + 2*x^3/3! + 12*x^5/5! + 128*x^7/7! + 1872*x^9/9! + ...
%e A302452 sinh(sinh(sinh(x))) = x/1! + 3*x^3/3! + 33*x^5/5! + 731*x^7/7! + 25857*x^9/9! + ...
%e A302452 sinh(sinh(sinh(sinh(x)))) = x/1! + 4*x^3/3! + 64*x^5/5! + 2160*x^7/7! + 121600*x^9/9! + ...
%e A302452 sinh(sinh(sinh(sinh(sinh(x))))) = x/1! + 5*x^3/3! + 105*x^5/5! + 4765*x^7/7! + 368145*x^9/9! + ...
%t A302452 Table[(2 n - 1)! SeriesCoefficient[Nest[Function[x, Sinh[x]], x, n], {x, 0, 2 n - 1}], {n, 15}]
%Y A302452 Cf. A003712, A003715.
%K A302452 nonn
%O A302452 1,2
%A A302452 _Ilya Gutkovskiy_, Apr 08 2018