E.g.f. S(x) = x + 2*x^3/3! + 24*x^5/5! + 872*x^7/7! + 67072*x^9/9! + 9174400*x^11/11! + 1999010432*x^13/13! + 644045742336*x^15/15! + 290850932891648*x^17/17! + ...
such that S(x) = Integral C(x) * C(S(x)) dx.
RELATED SERIES.
C(x) = 1 + x^2/2! + 5*x^4/4! + 109*x^6/6! + 5737*x^8/8! + 579961*x^10/10! + 98213933*x^12/12! + 25474555941*x^14/14! + 9505761607249*x^16/16! + 4872947687449969*x^18/18! + ...
such that C(x)^2 - S(x)^2 = 1.
C(x) + S(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 24*x^5/5! + 109*x^6/6! + 872*x^7/7! + 5737*x^8/8! + 67072*x^9/9! + 579961*x^10/10! + 9174400*x^11/11! + 98213933*x^12/12! + 1999010432*x^13/13! + 25474555941*x^14/14! + 644045742336*x^15/15! + 9505761607249*x^16/16! + 290850932891648*x^17/17! + 4872947687449969*x^18/18! + ...
such that C(x) + S(x) = exp( Integral C(S(x)) dx ).
C(S(x)) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 39929*x^8/8! + 5724249*x^10/10! + 1299323781*x^12/12! + 433635007877*x^14/14! + 201870080039537*x^16/16! + ...
S(S(x)) = x + 4*x^3/3! + 88*x^5/5! + 4992*x^7/7! + 549504*x^9/9! + 101239168*x^11/11! + 28464335360*x^13/13! + 11465663251456*x^15/15! + 6319308066455552*x^17/17! + ...
C(S(x)) + S(S(x)) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 88*x^5/5! + 493*x^6/6! + 4992*x^7/7! + 39929*x^8/8! + 549504*x^9/9! + 5724249*x^10/10! + 101239168*x^11/11! + 1299323781*x^12/12! + 28464335360*x^13/13! + 433635007877*x^14/14! + 11465663251456*x^15/15! + 201870080039537*x^16/16! + 6319308066455552*x^17/17! + ...
such that C(S(x)) + S(S(x)) = exp( Integral C(x) * C(S(x)) * C(S(S(x))) dx ).
If H(H(x)) = S(x) then
H(x) = x + x^3/3! + 7*x^5/5! + 205*x^7/7! + 13305*x^9/9! + 1616133*x^11/11! + 320304759*x^13/13! + 95177183745*x^15/15! + 40025542374641*x^17/17! + 22825140776633385*x^19/19! + 17079280074768716487*x^21/21! + 16337152342909182929909*x^23/23! + 19558206881883825876978857*x^25/25! + 28793090340440086848693036589*x^27/27! + 51357088945721875208166952420407*x^29/29! + ...
the nonzero coefficients of which appear to consist of only odd numbers.
|