# [seqfan] Re: A281181 - Need Formula for Terms

Paul Hanna pauldhanna.math at gmail.com
Fri Jan 20 03:12:56 CET 2017

```Franklin,
Somehow that link got messed up ... try:   https://oeis.org/A281181

Thanks!
Paul

On Thu, Jan 19, 2017 at 6:11 PM, Frank Adams-Watters <franktaw at netscape.net>
wrote:

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