%I #14 Feb 25 2019 17:58:46

%S 1,4,400,5364800,-367374176000,143449000888960000,

%T -181899009894595069440000,627436681283593072503040000000,

%U -5107564746905573153364013194240000000,88171417366157389105207649269976371200000000,-2969272543655823399308577388625291953035264000000000,182441297602875422577046590572630481727347923066880000000000

%N E.g.f. A(x) satisfies: A( tan( A( tanh(x) ) ) ) = x.

%C The series reversion of the e.g.f. is defined by A280793.

%H Paul D. Hanna, <a href="/A280791/b280791.txt">Table of n, a(n) for n = 1..50</a>

%F E.g.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3)/(4*n-3)! satisfies:

%F (1) A( tan( A( tanh(x) ) ) ) = x.

%F (2) A( tanh( A( tan(x) ) ) ) = x.

%F (3) tan( A( tanh( A(x) ) ) ) = x.

%F (4) tanh( A( tan( A(x) ) ) ) = x.

%F (5) A( tanh(A(x)) ) = arctan(x).

%F (6) A( tan(A(x)) ) = arctanh(x).

%F (7) Series_Reversion( A(x) ) = tan( A(tanh(x)) ) = tanh( A(tan(x)) ).

%e E.g.f.: A(x) = x + 4*x^5/5! + 400*x^9/9! + 5364800*x^13/13! - 367374176000*x^17/17! + 143449000888960000*x^21/21! - 181899009894595069440000*x^25/25! + 627436681283593072503040000000*x^29/29! - 5107564746905573153364013194240000000*x^33/33! + 88171417366157389105207649269976371200000000*x^37/37! - 2969272543655823399308577388625291953035264000000000*x^41/41! +...

%e such that A( tan( A( tanh(x) ) ) ) = x.

%e Note that A( A( tan( tanh(x) ) ) ) is NOT equal to x; 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 + 1/30*x^5 + 5/4536*x^9 + 479/555984*x^13 - 883111/855017856*x^17 + 1014203909/361219896576*x^21 - 5103375762413/435183970636800*x^25 + 77553540368447155/1092875131729446912*x^29 +...

%e RELATED SERIES.

%e A( tanh(x) ) = x - 2*x^3/3! + 20*x^5/5! - 552*x^7/7! + 29840*x^9/9! - 2520352*x^11/11! + 302768960*x^13/13! - 51218036352*x^15/15! + 12015036698880*x^17/17! - 3457794697175552*x^19/19! + 1042442536703513600*x^21/21! - 437297928076611069952*x^23/23! + 444983819928674567557120*x^25/25! +...

%e The series reversion of A( tanh(x) ) equals A( tan(x) ), which begins:

%e A( tan(x) ) = x + 2*x^3/3! + 20*x^5/5! + 552*x^7/7! + 29840*x^9/9! + 2520352*x^11/11! + 302768960*x^13/13! +...

%e tanh( A(x) ) = x - 2*x^3/3! + 20*x^5/5! - 440*x^7/7! + 16400*x^9/9! - 944800*x^11/11! + 82388800*x^13/13! - 9583600000*x^15/15! + 1041175200000*x^17/17! - 136472188736000*x^19/19! + 168221708270720000*x^21/21! - 77192574087699200000*x^23/23! - 152078345729585600000000*x^25/25! +...

%e The series reversion of tanh( A(x) ) equals tan( A(x) ), which begins:

%e tan( A(x) ) = x + 2*x^3/3! + 20*x^5/5! + 440*x^7/7! + 16400*x^9/9! + 944800*x^11/11! + 82388800*x^13/13! +...

%e The series reversion of A(x) = tan(A(tanh(x))) = tanh(A(tan(x))), and begins:

%e Series_Reversion( A(x) ) = x - 4*x^5/5! + 1616*x^9/9! - 10233664*x^13/13! + 605781862656*x^17/17! - 195074044306023424*x^21/21! + 226963189334487889924096*x^25/25! +...+ A280793(n)*x^(4*n-3)/(4*n-3)! +...

%o (PARI) {a(n) = my(A=x +x*O(x^(4*n+1))); for(i=1,2*n, A = A + (x - subst( tan(A) ,x, tanh(A) ) )/2; ); (4*n-3)!*polcoeff(A,4*n-3)}

%o for(n=1,20,print1(a(n),", "))

%Y Cf. A280790, A280792, A280793.

%K sign

%O 1,2

%A _Paul D. Hanna_, Jan 08 2017