A280791 E.g.f. A(x) satisfies: A( tan( A( tanh(x) ) ) ) = x.

1, 4, 400, 5364800, -367374176000, 143449000888960000, -181899009894595069440000, 627436681283593072503040000000, -5107564746905573153364013194240000000, 88171417366157389105207649269976371200000000, -2969272543655823399308577388625291953035264000000000, 182441297602875422577046590572630481727347923066880000000000

Offset

1,2

Comments

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

Links

Paul D. Hanna, Table of n, a(n) for n = 1..50

Formula

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

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

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

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

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

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

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

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

Example

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! +...
such that A( tan( A( tanh(x) ) ) ) = x.
Note that A( A( tan( tanh(x) ) ) ) is NOT equal to x; the composition of these functions is not commutative.
The e.g.f. as a series with reduced fractional coefficients begins:
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 +...
RELATED SERIES.
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! +...
The series reversion of A( tanh(x) ) equals A( tan(x) ), which begins:
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! +...
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! +...
The series reversion of tanh( A(x) ) equals tan( A(x) ), which begins:
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! +...
The series reversion of A(x) = tan(A(tanh(x))) = tanh(A(tan(x))), and begins:
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)! +...

Prog

(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)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A280790, A280792, A280793.

Sequence in context: A349460 A158111 A259049 * A198709 A326209 A287965

Adjacent sequences: A280788 A280789 A280790 * A280792 A280793 A280794

Keyword

sign

Author

Paul D. Hanna, Jan 08 2017

Status

approved