A279836 E.g.f. A(x) satisfies: A( sin( A(x) ) ) = sinh(x).

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

Offset: 1

Author

Paul D. Hanna, Jan 11 2017

Keywords

• sign

Comments

First negative term is a(75), the coefficient of x^149 in A(x).

Apart from signs, essentially the same terms as A279838.

Examples

			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( sin( A(x) ) ) = sinh(x).
Note that A(A(x)) is NOT equal to sinh(arcsin(x)) nor arcsin(sinh(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 + ...
RELATED SERIES.
A( sin(x) ) = x - 4*x^5/5! + 28*x^7/7! - 976*x^9/9! + 38016*x^11/11! - 3272736*x^13/13! + 321487680*x^15/15! - 47598285056*x^17/17! + 8350711540224*x^19/19! - 1819783398735872*x^21/21! + ...
The series reversion of A( sin(x) ) equals A( arcsinh(x) ), which begins:
A( arcsinh(x) ) = x + 4*x^5/5! - 28*x^7/7! + 2992*x^9/9! - 126720*x^11/11! + 20505952*x^13/13! - 2396136256*x^15/15! + ...
sin( A(x) ) = x - 4*x^5/5! - 28*x^7/7! - 976*x^9/9! - 38016*x^11/11! - 3272736*x^13/13! - 321487680*x^15/15! - 47598285056*x^17/17! - 8350711540224*x^19/19! - 1819783398735872*x^21/21! + ...
The series reversion of sin( A(x) ) equals arcsinh( A(x) ), which begins:
arcsinh( A(x) ) = x + 4*x^5/5! + 28*x^7/7! + 2992*x^9/9! + 126720*x^11/11! + 20505952*x^13/13! + 2396136256*x^15/15! + ...
The series reversion of A(x) = sin(A(arcsinh(x))) = arcsinh(A(sin(x))), and begins:
Series_Reversion( 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! + ...
		

Links

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

Crossrefs

Cf. A279838, A280790, A280792, A279837.

Programs

PARI

{a(n) = my(X = x +x*O(x^(2*n)),A=X); for(i=1, 2*n, A = A + (sinh(X) - subst(A,x, sin(A) ) )/2; H=A ); (2*n-1)!*polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))

Formula

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

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

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

(3) arcsinh( A( sin( A(x) ) ) ) = x.

(4) sin( A( arcsinh( A(x) ) ) ) = x.

(5) A( sin( A( arcsinh(x) ) ) ) = x.

(6) A( arcsinh( A( sin(x) ) ) ) = x.

(7) Series_Reversion( A(x) ) = sin( A( arcsinh(x) ) ) = arcsinh( A( sin(x) ) ), and equals the e.g.f. of A279838.