[seqfan] Re: A281181 - Need Formula for Terms

[Frank Adams-Watters]

Is that supposed to be A281183? I don't find any sequence A281181.

Franklin T. Adams-Watters


-----Original Message-----
From: Paul Hanna <pauldhanna.math at gmail.com>
To: seqfan <seqfan at list.seqfan.eu>
Sent: Thu, Jan 19, 2017 5:02 pm
Subject: [seqfan] A281181 - Need Formula for Terms

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 nicesurprise:(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--Seqfan Mailing list - http://list.seqfan.eu/