[seqfan] Re: Generating function for A000992

Paul D Hanna pauldhanna at juno.com
Wed Dec 21 00:18:28 CET 2011

As you may have noticed, the relations I provided are not explicit. 
The relations below are satisfied by the g.f. A(x) of A000992, 
but unfortunately they do not isolate A(x).   
They are pretty (to me), but too symmetric; reversing the signs produces the same formulas. 
Perhaps someone else could find an explicit expression for A(x) ... 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Generating function for A000992
Date: Tue, 20 Dec 2011 21:01:52 GMT

There may be a more compact g.f. for A000992, but here is one approach. 

Let A(x) = Sum_{n>=0} A000992(n) * x^n, then 

1 = (A(x) + A(-x))/2 * (1 - x*(A(x) - A(-x))/2) 

which is equivalent to: 

-2 + 2*A(x) - x*A(x)^2  =  2 - 2*A(-x) - x*A(-x)^2 

= Sum_{n>=0} A000992(n)^2 * x^(2*n+1) 
= x + x^3 + x^5 + 4*x^7 + 9*x^9 + 36*x^11 + 121*x^13 + 576*x^15 + 2209*x^17 +... 

an odd function with coefficients equal to the squares of A000992(n). 

- Paul D. Hanna

---------- Original Message ----------
From: "N. J. A. Sloane" <njas at research.att.com>
To: seqfan at seqfan.eu
Cc: njas at research.att.com
Subject: [seqfan] Generating function for A000992
Date: Tue, 20 Dec 2011 11:09:22 -0500

Dear Sequence Fans,
A correspondent asks if there is a formula or g.f. for A000992:

> Date: Tue, 20 Dec 2011 12:50:41 +0100
> From: Jan Schwientek <Jan.Schwientek at itwm.fraunhofer.de>
> I'm PhD student and interested in an explicit representation and/or
> generating function for the integer sequence A000992

Can anyone help him?


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