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a(n) = coefficient of x^(2*n-1) in the n-th iteration (n-fold self-composition) of e.g.f. tan(x).
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%I #4 Apr 09 2018 22:33:48

%S 1,4,168,26400,10373760,8226518272,11524607732736,26047611675267072,

%T 88935060882222120960,436394080487109570265088,

%U 2959343413232671759344861184,26874522377891724867898947141632,318464577992023576681854032513335296,4818779071094868918454887699722367139840

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

%C 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).

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%e The initial coefficients of successive iterations of e.g.f. A(x) = tan(x) (odd powers only) are as follows:

%e n = 1: (1), 2, 16, 272, 7936, ... e.g.f. A(x)

%e n = 2: 1, (4), 72, 2896, 203904, ... e.g.f. A(A(x))

%e n = 3: 1, 6, (168), 10672, 1198080, ... e.g.f. A(A(A(x)))

%e n = 4: 1, 8, 304, (26400), 4071040, ... e.g.f. A(A(A(A(x))))

%e n = 5: 1, 10, 480, 52880, (10373760), ... e.g.f. A(A(A(A(A(x)))))

%e ...

%e More explicitly, the successive iterations of e.g.f. A(x) = tan(x) begin:

%e tan(x) = x/1! + 2*x^3/3! + 16*x^5/5! + 272*x^7/7! + 7936*x^9/9! + ...

%e tan(tan(x)) = x/1! + 4*x^3/3! + 72*x^5/5! + 2896*x^7/7! + 203904*x^9/9! + ...

%e tan(tan(tan(x))) = x/1! + 6*x^3/3! + 168*x^5/5! + 10672*x^7/7! + 1198080*x^9/9! + ...

%e tan(tan(tan(tan(x)))) = x/1! + 8*x^3/3! + 304*x^5/5! + 26400*x^7/7! + 4071040*x^9/9! + ...

%e tan(tan(tan(tan(tan(x))))) = x/1! + 10*x^3/3! + 480*x^5/5! + 52880*x^7/7! + 10373760*x^9/9! + ...

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

%Y Cf. A000182, A003718, A003720.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Apr 08 2018