OFFSET
1,2
COMMENTS
The series reversion of the e.g.f. is defined by A280792.
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( sin( A( sinh(x) ) ) ) = x.
(2) A( sinh( A( sin(x) ) ) ) = x.
(3) sin( A( sinh( A(x) ) ) ) = x.
(4) sinh( A( sin( A(x) ) ) ) = x.
(5) A( sinh(A(x)) ) = asin(x).
(6) A( sin(A(x)) ) = asinh(x).
(7) Series_Reversion( A(x) ) = sin( A(sinh(x)) ) = sinh( A(sin(x)) ).
EXAMPLE
E.g.f.: A(x) = x + 4*x^5/5! + 2320*x^9/9! + 9857600*x^13/13! + 159122080000*x^17/17! + 7098806416000000*x^21/21! + 686863244097538560000*x^25/25! + 143579312211740504320000000*x^29/29! + 27634174819420517051458560000000*x^33/33! + 103635121107833144489335056076800000000*x^37/37! - 624322694794393812097710416148436992000000000*x^41/41! +...
such that A( sin( A( sinh(x) ) ) ) = x.
Note that A( A( sin( sinh(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 + 29/4536*x^9 + 6161/3891888*x^13 + 382505/855017856*x^17 + 50189525/361219896576*x^21 + 134894899309/3046287794457600*x^25 + 195216389950265/12021626449023916032*x^29 + ...
RELATED SERIES.
A( sinh(x) ) = x + x^3/3! + 5*x^5/5! + 141*x^7/7! + 6185*x^9/9! + 482681*x^11/11! + 55181165*x^13/13! + 8650849221*x^15/15! + 1806577140945*x^17/17! + 482615036315761*x^19/19! + 160833575943581525*x^21/21! + 65507016886932658301*x^23/23! + 32006289578900322278905*x^25/25! + ...
The series reversion of A( sinh(x) ) equals A( sin(x) ), which begins:
A( sin(x) ) = x - x^3/3! + 5*x^5/5! - 141*x^7/7! + 6185*x^9/9! - 482681*x^11/11! + 55181165*x^13/13! + ...
sinh( A(x) ) = x + x^3/3! + 5*x^5/5! + 85*x^7/7! + 2825*x^9/9! + 151625*x^11/11! + 12098125*x^13/13! + 1339476125*x^15/15! + 196410020625*x^17/17! + 37062144900625*x^19/19! + 8772471210303125*x^21/21! + 2519410212081953125*x^23/23! + 854580849916226265625*x^25/25! + ... + A318635(n)*x^(2*n-1)/(2*n-1)! + ...
The series reversion of sinh( A(x) ) equals sin( A(x) ), which begins:
sin( A(x) ) = x - x^3/3! + 5*x^5/5! - 85*x^7/7! + 2825*x^9/9! - 151625*x^11/11! + 12098125*x^13/13! + ...
The series reversion of A(x) = sin(A(sinh(x))) = sinh(A(sin(x))), and begins:
Series_Reversion( A(x) ) = x - 4*x^5/5! - 304*x^9/9! + 648896*x^13/13! + 2650020096*x^17/17! - 142483330376704*x^21/21! + 24311838501965418496*x^25/25! +...+ A280792(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( sin(A) , x, sinh(A) ) )/2; H=A ); (4*n-3)!*polcoeff(A, 4*n-3)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 08 2017
STATUS
approved