[seqfan] Re: G.F. for A245925 Sought

[Paul D Hanna]

Seqfans, 
 In general, we have the binomial identity: 
  
if   b(n) = Sum_{k=0..n}  t^k * C(2*k, k) * C(n+k, n-k), 
  
then  b(n)^2 = Sum_{k=0..n}  (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k).    .
Other examples: 
 https://oeis.org/A243943
 https://oeis.org/A243944 https://oeis.org/A243007 
  
In case anyone is interested. 
   Paul 
 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: G.F. for A245925 Sought
Date: Mon, 18 Aug 2014 02:17:50 GMT

Seqfans,  
     Here is very similar sequence that needs a generating function: https://oeis.org/A243945  
defined term-wise by: 
   a(n) = Sum_{k=0..n} C(2*k, k)^2 * C(n+k, n-k). 
   
That binomial sum is similar to Peter Luschny's formula (and Robert Israel, modified) for A245925: 
   A245925(n) = (-1)^n * Sum_{k=0..n} C(2*k, k) * C(n+k, n-k)^2. 
  
And as before, the bisections of A243945 involve perfect squares:     a(2*n) = A243946(n)^2,  
   a(2*n+1) = 5 * A243947(n)^2. 
 The g.f.s formed from a(2*n)^(1/2) and (a(2*n+1)/5)^(1/2) are: 
A243946: sqrt( (1+x + sqrt(1-18*x+x^2)) / (2*(1-18*x+x^2)) ); 
A243947: sqrt( (1+x - sqrt(1-18*x+x^2)) / (10*x*(1-18*x+x^2)) ). 
 Perhaps someone can find a g.f. for this sequence ...  
  I would like to thank Peter Luschny and Vaclav Kotesovec for their valuable contributions 
to the sequences mentioned in the prior email.  
Thanks, 
    Paul 
 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: G.F. for A245925 Sought
Date: Sat, 16 Aug 2014 16:32:19 GMT


Seqfans,     Here is an interesting sequence that needs a g.f. with a closed form: 

http://oeis.org/A245925 
1, -3, 25, -243, 2601, -29403, 344569, -4141875, 50737129, ... 
  
The generating function is given by the binomial series identity: 
 
A(x) = Sum_{n>=0} x^n*Sum_{k=0..n} (-1)^k * C(n,k)^2 * Sum_{j=0..k} C(k,j)^2 * x^j 

= Sum_{n>=0} x^n / (1+x)^(2*n+1) * [ Sum_{k=0..n} C(n,k)^2*(-x)^k ]^2  

and the term-by-term formula is: 

a(n) = Sum_{k=0..n} Sum_{j=0..2*n-2*k} (-1)^(j+k) * C(2*n-k,j+k)^2 * C(j+k,k)^2. 


What is surprising about the terms in A245925 is that they involve perfect squares: 

a(2*n) = A245926(n)^2, 

a(2*n+1) = (-3)*A245927(n)^2, 


where the g.f.s for A245926 and A245927 have a closed form: 

A245926: sqrt( (1-x + sqrt(1-14*x+x^2)) / (2*(1-14*x+x^2)) ), 

A245927: sqrt( (1-x - sqrt(1-14*x+x^2)) / (6*x*(1-14*x+x^2)) ). 


Perhaps someone could derive a g.f. with a ~similar closed form for A245925. 

Thanks, 
   Paul

_______________________________________________

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