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