[seqfan] A281181 - Need Formula for Terms

[Paul Hanna]

Seqfans,
    Let C(x) be the e.g.f. described by  https://oeis.org/A281181,
then we have these beautiful results:

(1) C(x)^1 = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ),

(2) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2      dx ),

(3) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3   dx ),

(4) C(x)^4 = d/dx Series_Reversion( Integral 1/(1 + x^2)^2 dx ).

I have not found a simple formula for C(x)^n for n>4, but here is a nice
surprise:

(5) C(x)^5 = d/dx Series_Reversion( Integral C(i*x)^5    dx ).

[Note that an infinite number of functions satisfies condition (5).]


A function this lovely must have a nice formula for the coefficients,
and surely there is a combinatorial interpretation yet to be divulged.

Can anyone find a formula for the terms in A281181?
(That might be asking too much, but I had to ask.)

Thanks,
     Paul