A279838 E.g.f. A(x) satisfies: A( sinh( A(x) ) ) = sin(x).

1, -1, 5, -113, 4505, -324545, 34312317, -5171466801, 1036525185393, -268061777199361, 86654517306871861, -34236056076864607345, 16224034929841344607625, -9077085568599515191480769, 5918716657866577845713460525, -4447229534037550877037585953073, 3813957492790787345317821024498657, -3702048025219670721125627874960351233

Offset

1,3

Comments

Apart from signs, essentially the same terms as A279836.

Links

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

Formula

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

(1) A( sinh( A(x) ) ) = sin(x).

(2) A( arcsin( A(x) ) ) = arcsinh(x).

(3) arcsin( A( sinh( A(x) ) ) ) = x.

(4) sinh( A( arcsin( A(x) ) ) ) = x.

(5) A( sinh( A( arcsin(x) ) ) ) = x.

(6) A( arcsin( A( sinh(x) ) ) ) = x.

(7) Series_Reversion( A(x) ) = sinh( A( arcsin(x) ) ) = arcsin( A( sinh(x) ) ), and equals the e.g.f. of A279836.

Example

E.g.f.: A(x) = x - x^3/3! + 5*x^5/5! - 113*x^7/7! + 4505*x^9/9! - 324545*x^11/11! + 34312317*x^13/13! - 5171466801*x^15/15! + 1036525185393*x^17/17! - 268061777199361*x^19/19! + 86654517306871861*x^21/21! - 34236056076864607345*x^23/23! + 16224034929841344607625*x^25/25! + ...
such that A( sinh( A(x) ) ) = sin(x).
Note that A(A(x)) is NOT equal to sin(arcsinh(x)) nor arcsinh(sin(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 - (1/6)*x^3 + (1/24)*x^5 - (113/5040)*x^7 + (901/72576)*x^9 - (64909/7983360)*x^11 + (879803/159667200)*x^13 - (1723822267/435891456000)*x^15 + ...

Prog

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

Crossrefs

Cf. A279836, A280790, A280792, A279839.

Sequence in context: A241109 A208317 A279836 * A009603 A009678 A009691

Adjacent sequences: A279835 A279836 A279837 * A279839 A279840 A279841

Keyword

sign

Author

Paul D. Hanna, Jan 11 2017

Status

approved