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%I
%S 1,2,20,496,23120,1747360,195269568,30288321792,6227935871232,
%T 1639388975800832,537520438716580864,214739554795652526080,
%U 102653241459277667225600,57838071113129054500200448,37921092324167375349735014400,28616681138798042948070311264256,24621851021674983535130840611749888,23955560260216279396643234915721281536
%N E.g.f. A(x) satisfies: A( tanh( A(x) ) ) = tan(x).
%C First negative term is a(65), the coefficient of x^129 in A(x).
%C Apart from signs, essentially the same terms as A279839.
%H Paul D. Hanna, <a href="/A279837/b279837.txt">Table of n, a(n) for n = 1..100</a>
%F E.g.f. A(x) satisfies:
%F (1) A( tanh( A(x) ) ) = tan(x).
%F (2) A( arctan( A(x) ) ) = arctanh(x).
%F (3) arctan( A( tanh( A(x) ) ) ) = x.
%F (4) tanh( A( arctan( A(x) ) ) ) = x.
%F (5) A( tanh( A( arctan(x) ) ) ) = x.
%F (6) A( arctan( A( tanh(x) ) ) ) = x.
%F (7) Series_Reversion( A(x) ) = tanh( A( arctan(x) ) ) = arctan( A( tanh(x) ) ), and equals the e.g.f. of A279839.
%e E.g.f.: A(x) = x + 2*x^3/3! + 20*x^5/5! + 496*x^7/7! + 23120*x^9/9! + 1747360*x^11/11! + 195269568*x^13/13! + 30288321792*x^15/15! + 6227935871232*x^17/17! + 1639388975800832*x^19/19! + 537520438716580864*x^21/21! + 214739554795652526080*x^23/23! + 102653241459277667225600*x^25/25! + ...
%e such that A( tanh( A(x) ) ) = tan(x).
%e Note that A(A(x)) is NOT equal to tan(arctanh(x)) nor arctanh(tan(x)) since the composition of these functions is not commutative.
%e The e.g.f. as a series with reduced fractional coefficients begins:
%e A(x) = x + 1/3*x^3 + 1/6*x^5 + 31/315*x^7 + 289/4536*x^9 + 10921/249480*x^11 + 78233/2494800*x^13 + 4381991/189189000*x^15 + ...
%o (PARI) {a(n) = my(X = x +x*O(x^(2*n)),A=X); for(i=1, 2*n, A = A + (tan(X) - subst(A,x, tanh(A) ) )/2; H=A ); (2*n-1)!*polcoeff(A, 2*n-1)}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A279839, A280791, A280793, A279836.
%K sign
%O 1,2
%A _Paul D. Hanna_, Jan 11 2017
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