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A279837 E.g.f. A(x) satisfies: A( tanh( A(x) ) ) = tan(x). 3
1, 2, 20, 496, 23120, 1747360, 195269568, 30288321792, 6227935871232, 1639388975800832, 537520438716580864, 214739554795652526080, 102653241459277667225600, 57838071113129054500200448, 37921092324167375349735014400, 28616681138798042948070311264256, 24621851021674983535130840611749888, 23955560260216279396643234915721281536 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First negative term is a(65), the coefficient of x^129 in A(x).

Apart from signs, essentially the same terms as A279839.

LINKS

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

FORMULA

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

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

(2) A( arctan( A(x) ) ) = arctanh(x).

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

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

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

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

(7) Series_Reversion( A(x) ) = tanh( A( arctan(x) ) ) = arctan( A( tanh(x) ) ), and equals the e.g.f. of A279839.

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

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.

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 + (tan(X) - subst(A, x, tanh(A) ) )/2; H=A ); (2*n-1)!*polcoeff(A, 2*n-1)}

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

CROSSREFS

Cf. A279839, A280791, A280793, A279836.

Sequence in context: A012816 A012340 A279839 * A279200 A277414 A296789

Adjacent sequences: A279834 A279835 A279836 * A279838 A279839 A279840

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 11 2017

STATUS

approved

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Last modified January 28 14:40 EST 2023. Contains 359895 sequences. (Running on oeis4.)