[seqfan] A000012 Generating Function

franktaw at netscape.net franktaw at netscape.net
Mon Jun 29 22:21:11 CEST 2009

A few years ago I added generating functions for A000012 (the all 1's 
sequence) regarded as an array.  The one case I did not include was the 
doubly exponential generating function regarding the sequence as a 
triangular array; i.e.

f(x,y) = sum_{0 <= k <= n} (x^k y^n) / (k! n!).

I'm wondering if anyone can give an explicit form for this.

Note that

f(x,y) + f(y,x) - e^{x+y} = I_0(2*sqrt(x*y)),

where I is the modified Bessel function of the first kind; so any 
solution is going to have involve functions at least as exotic as this 
Bessel function.

This is probably one of those things that is well-known by those who 
know such things.

Franklin T. Adams-Watters

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