G.f. for C(q^n,n)? - Generalized

Sun Jan 13 06:14:21 CET 2008

Seqfans, 
     There is much more to this study that needs to be explored. 
Here is my most general formula so far: 

(9) Let F,G,H, be formal power series in x such that F(0)=1, G(0)=1, 
then 
Sum_{n>=0} m^n * H(q^n*x) * log( F(q^n*x)*G(x) )^n / n! = 
Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] H(y)*F(y)^(m*q^n). 

But there are yet many undiscovered formulas for sums of this type, 
such as these examples: 
(10) Sum_{n>=0} log((1 + a*p^n*x)*(1 + b*q^n*x))^n/n!  = ? 
(11) Sum_{n>=0} log(1 + (a*p^n + b*q^n)*x)^n/n!  = ? 
both yield series with integer coefficients whenever p,q,a,b, are
integer. 
These lead to much more general sums that seem impossible to find a
formula for. 
Yet it intrigues me that these sums yield integer series for integer
arguments.  

Further, not just an exponential sum of powers of logs can be studied; 
it appears that other functions with their inverses also yield integer
series. 

Here is an example using the hyperbolic sine series applied on the
inverse sinh: 

(12) G.f.: A(x) = Sum_{n>=0} asinh( 2^(2n+1)*x )^(2n+1) / (2n+1)!  = 
2*x + 84*x^3 + 276892*x^5 + 111457917800*x^7 + 
6660816097416169260*x^9 + 66597307693046550483175282456*x^11 +
120167520447600665027319450022840022638104*x^13 +...

Is there a simple formula for the integer coefficients on the right side
of (12)? 

Of course (12) is an example of the more general: 
(13) G.f.: A(x) = Sum_{n>=0} asinh( q^(2n+1)*x )^(2n+1) / (2n+1)!  = ? 
yielding some unknown integer series for all integer q. 

Is there a simple formula for the integer coefficients on the right side
of (13)? 
It would be nice if it turned out to be as simple a formula as: 
(2) Sum_{n>=0} log(1 + q^n*x)^n/n!  =  Sum_{n>=0} C(q^n,n)*x^n. 

What other functions can be used besides  exp(), sinh(), etc., 
in a similar nontrivial sum that yields an integer series for all integer
q? 

Unless I find a breakthrough, this will be the last I post regarding
these sums.  

But I would like very much to hear from anyone with answers to the above 
questions or a nice application of any of these sums. 

Thanks, 
     Paul 

On Tue, 8 Jan 2008 22:50:04 -0500 pauldhanna at juno.com writes:
SeqFans, 
     A little more on the generalization. 
Introduce a function G(x) (independent of summation index n): 

Let F(x) be any formal power series in x such that F(0)=1; then 
(7) Sum_{n>=0} m^n * F(q^n*x)^b * log( G(x)*F(q^n*x) )^n / n!  =  
     Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] F(y)^(m*q^n + b) 

where [y^n] F(y) denotes the coefficient of y^n in F(y).

It seems counter-intuitive that the power of G would be m*q^n; 
however, it can be deduced from (5) (in prior email below). 

EXAMPLE of (7).   
When applying (7) to A136554:  
1,3,10,82,2304,232088,81639942,99425060368,421915147527984,
G.f.: A(x) = Sum_{n>=0} log( (1 + x)*(1 + 2^n*x) )^n / n!.

it reveals the nice formulas:

G.f.: A(x) = Sum_{n>=0} C(2^n,n) * x^n * (1+x)^(2^n). 

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

So, from (7) we find a nice simplification for: 
G.f.: A(x) = Sum_{n>=0} log( z*(1 + 2^n*x) )^n / n!
as:  
G.f.: A(x) = Sum_{n>=0} C(2^n,n) * x^n * z^(2^n)

FURTHER YET ... 
But here is where I get stumped. 
Can the following sum be simplified into a form similar to (7)?  

(8) Sum_{n>=0} log( F(p^n*x)*G(q^n*x) )^n / n!  = ? 

This sum again returns an integer series in x when both 
F(x) and G(x) are integer series in x for all integer p, q. 

But simplifying (8) into a form like (7) when p not= q 
is not straitforward (at least for me). 

Perhaps someone can find a nice form for (8)? 

EXAMPLE of (8).
Let p=2, q=3, then we have A136578:
G.f.: A(x) = Sum_{n>=0} log( (1 + 2^n*x)*(1 + 3^n*x) )^n / n!.
1,5,78,6527,3450452,12594729052,338284182093366,70004091118158663618,

But I have not found a nice formula for even this simple case ... 
     Paul 

On Wed, 2 Jan 2008 21:50:02 -0500 pauldhanna at juno.com writes:
Seqfans,
     The identity can be generalized further. 
Let F(x) be any formal power series in x such that F(0)=1. 
Then 
(5) Sum_{n>=0} m^n * F(q^n*x)^b * log( F(q^n*x) )^n / n!  =  
     Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) 

where [y^n] G(y) denotes the coefficient of y^n in G(y).

If we let F(x) = exp(x), then we have the nontrivial result 

(6) Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n!  =  
     Sum_{n>=0} (m*q^n + b)^n * x^n / n! 

which becomes trivial at b=0. 
Example of (6): q=2, m=1, b=1:
exp(x) + 2*exp(2x) + 2^4*exp(4x)*x^2/2! + 2^9*exp(8x)*x^3/3! +...+
   2^(n^2)*exp(2^n*x)/n! +...
= 1 + 3x + 5^2*x^2/2! + 9^3*x^3/3! + 17^4*x^4/4! +...+ 
   (2^n+1)^n*x^n/n! +... 

Example of (6): q=2, m=1, b=-1:
exp(-x) + 2*exp(-2x) + 2^4*exp(-4x)*x^2/2! + 2^9*exp(-8x)*x^3/3! +...+
   2^(n^2)*exp(-2^n*x)/n! +...
= 1 + x + 3^2*x^2/2! + 7^3*x^3/3! + 15^4*x^4/4! +...+ 
   (2^n-1)^n*x^n/n! +...

Example of (5): F(x) = 1+x+x^2, q=2, m=1, b=0: 
Sum_{n>=0} log( 1 + 2^n*x + 2^(2n)*x^2 )^n / n!  =  
    Sum_{n>=0} T(2^n, n) * x^n  
where T(2^n,n) = trinomial coefficient of x^n in (1+x+x^2)^(2^n).  

And many other examples can be given. 

So the question arises, does (5) have useful applications? 
Does it offer a g.f. for some significant sequences in OEIS? 
     Paul 

On Sun, 30 Dec 2007 21:03:39 -0500 pauldhanna at juno.com writes:
Seqfans, 
     Recall the identity: 
(2) Sum_{n>=0} log(1 + q^n*x)^n/n!  =  Sum_{n>=0} C(q^n,n)*x^n. 

 From this, I found the more general statements: 

(3) Sum_{n>=0} m^n*log(1 + q^n*x)^n/n!  =  Sum_{n>=0} C(m*q^n,n)*x^n.

(4) Sum_{n>=0} m^n * (1 + q^n*x)^b * log(1 + q^n*x)^n/n!  =  
     Sum_{n>=0} C(m*q^n + b, n)*x^n.

Identity (4) is very interesting ... I wonder if it leads to other
results? 
It certainly can lead to many significant sequences! 

What I would really like is for formula (4) to allow the g.f. 
    A(x,m,b) = Sum_{n>=0} C(m*q^n + b, n)*x^n 
to be manipulated to solve some functional equation ... 

Any ideas along these lines from anyone?
      Paul
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