%I #16 Sep 05 2018 20:40:44

%S 1,-4,-304,648896,2650020096,-142483330376704,24311838501965418496,

%T -17572131142184492046434304,31550058162566932127305417424896,

%U -123841868587916789535717370523560443904,969729634851676570691527174556498457233719296,-14068736567241332813708145418894026558391075423125504,356436464229966658550949874743523835716465340767523041181696,-15023108679681039882374036580197265042861509571919315150655773999104

%N E.g.f. A(x) satisfies: A( arcsin( A( arcsinh(x) ) ) ) = x.

%C The series reversion of the e.g.f. is defined by A280790.

%H Paul D. Hanna, <a href="/A280792/b280792.txt">Table of n, a(n) for n = 1..50</a>

%F E.g.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3)/(4*n-3)! satisfies:

%F (1) A( arcsin( A( arcsinh(x) ) ) ) = x.

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

%F (3) arcsin( A( arcsinh( A(x) ) ) ) = x.

%F (4) arcsinh( A( arcsin( A(x) ) ) ) = x.

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

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

%F (7) Series_Reversion( A(x) ) = arcsin( A(arcsinh(x)) ) = arcsinh( A(arcsin(x)) ).

%e E.g.f.: 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! - 17572131142184492046434304*x^29/29! + 31550058162566932127305417424896*x^33/33! - 123841868587916789535717370523560443904*x^37/37! + 969729634851676570691527174556498457233719296*x^41/41! + ...

%e such that A( arcsin( A( arcsinh(x) ) ) ) = x.

%e Note that A( A( arcsin( arcsinh(x) ) ) ) is NOT equal to x; the composition of these functions is not commutative.

%e The e.g.f. as a series with reduced fractional coefficients begins:

%e A(x) = x - 1/30*x^5 - 19/22680*x^9 + 10139/97297200*x^13 + 3450547/463134672000*x^17 - 139143877321/49893498214560000*x^21 + 5935507446768901/3786916514485104000000*x^25 - 4413653374109964767/2220816151494708768000000*x^29 + ...

%e RELATED SERIES.

%e A( arcsinh(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! + ...

%e The series reversion of A( arcsinh(x) ) equals A( arcsin(x) ), which begins:

%e A( arcsin(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! + ... + A318635(n)*x^(2*n-1)/(2*n-1)! + ...

%e arcsinh( A(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! + ...

%e The series reversion of arcsinh( A(x) ) equals arcsin( A(x) ), which begins:

%e arcsin( A(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! + ...

%e The series reversion of A(x) begins:

%e Series_Reversion( 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! +...+ A280790(n)*x^(4*n-3)/(4*n-3)! +...

%o (PARI) {a(n) = my(A=x +x*O(x^(4*n+1))); for(i=1,2*n, A = A + (x - subst( asin(A) ,x, asinh(A) ) )/2; H=A ); (4*n-3)!*polcoeff(A,4*n-3)}

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

%Y Cf. A280790, A280791, A280793, A318635.

%K sign

%O 1,2

%A _Paul D. Hanna_, Jan 09 2017