A279839 E.g.f. A(x) satisfies: A( tan( A(x) ) ) = tanh(x).

1, -2, 20, -496, 23120, -1747360, 195269568, -30288321792, 6227935871232, -1639388975800832, 537520438716580864, -214739554795652526080, 102653241459277667225600, -57838071113129054500200448, 37921092324167375349735014400, -28616681138798042948070311264256, 24621851021674983535130840611749888

Offset

1,2

Comments

Apart from signs, essentially the same terms as A279837.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Formula

E.g.f. A(x) satisfies:

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

(2) A( atanh( A(x) ) ) = atan(x).

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

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

(5) A( tan( A( atanh(x) ) ) ) = x.

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

(7) Series_Reversion( A(x) ) = tan( A( atanh(x) ) ) = atanh( A( tan(x) ) ), and equals the e.g.f. of A279837.

Example

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! +...
such that A( tan( A(x) ) ) = tanh(x).
Note that A(A(x)) is NOT equal to tanh(atan(x)) nor atan(tanh(x)) since the composition of these functions is not commutative.
The e.g.f. as a series with reduced fractional coefficients begins:
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 +...

Prog

(PARI) {a(n) = my(X = x +x*O(x^(2*n)), A=X); for(i=1, 2*n, A = A + (tanh(X) - subst(A, x, tan(A) ) )/2; H=A ); (2*n-1)!*polcoeff(A, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A279837, A280791, A280793, A279838.

Sequence in context: A352250 A012816 A012340 * A279837 A279200 A277414

Adjacent sequences: A279836 A279837 A279838 * A279840 A279841 A279842

Keyword

sign

Author

Paul D. Hanna, Jan 11 2017

Status

approved