[seqfan] A281181 - Need Formula for Terms
Paul Hanna
pauldhanna.math at gmail.com
Fri Jan 20 00:01:55 CET 2017
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
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