[seqfan] Re: Surprising Patterns in Tangent and Secant Numbers

[Max Alekseyev]

It appears that 1/cosh(x) can be replaced by an arbitrary function g(x):

serreverse(x*serlaplace(exp(x)*g(x))) = 1/( 1 +
1/serreverse(x*serlaplace(g(x))) )

Max

On Tue, Nov 10, 2009 at 2:23 AM, Paul D Hanna <pauldhanna at juno.com> wrote:
> SeqFans,
>     I suppose that the proofs of the main observations in my prior email
> could start with establishing the following identity (in PARI notation):
>
> x/serreverse(x*serlaplace(exp(x)/cosh(x)))
>
> = x + x/serreverse(x*serlaplace(1/cosh(x)))
>
> = 1 + x - x^2 + 3*x^4 - 38*x^6 + 947*x^8 - 37394*x^10 +...
>
> Is this identity easy to prove?
>      Paul
>
> Cf. A157308.
>
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