[seqfan] Re: Another possible generating function for A066318.

[israel at math.ubc.ca]

There are a few things not quite right here.  My integrand was

cos(t)*W(t+2*k*Pi)/((t+2*k*Pi)*(1+W(t+2*k*Pi)))

(note the * rather than + in the denominator).

The asymptotic series appears to be 

-sum_{n=1}^\infty  W(2*k*Pi)^(2*n+1) * P_n(W(2*k*Pi)) * Q_n(Pi^2)/
(2^(2*n) * Pi^(2*n) * k^(2*n+1) * (W(2*k*Pi)+1)^(4*n+1))

where P_n is an integer polynomial of degree 2*n and 
Q_n(t) = sum_{m=0}^{n-1} (-1)^(m+n-1)*t^m/(2*m+1)!
  
The coefficients of P_n appear to form the (2n+1)'th row of A042977.
For example, 
P_3(w) = 720*w^6+8640*w^5+45756*w^4+137512*w^3+248250*w^2+255828*w+117649.
I suppose this is quite plausible given that A042977 comes from the 
derivatives of LambertW, but I don't have a proof.  

On the other hand, A066318(n) is just (n-1)!*2^n, so I wouldn't think of
this as having much to do with A066318.

Robert Israel
University of British Columbia



On Sep 2 2012, Ed Jeffery wrote:

>The following involves sequences A066318 [1] and A042977 [2]:
>
>Let S_{a}^{b} f(x)*dx denote integration of f(x) from a to b. Let W()
>denote the Lambert W-function. When the integral
>
>(1)   S_{-Pi}^{Pi} [cos(t)*W(t+2*k*Pi)/((t+2*k*Pi)+(1+W(t+2*k*Pi)))]*dt
>
>is expanded to an asymptotic series in k followed by term-by-term
>integration, as specified by Robert Isreal in [3], it appears that the
>resulting series could be written as
>
> (2) Sum_{n=1,2,...} 
> {[(1/A066318(n))*(cos(t)*W(2*k*Pi)^n*t^(n-1))/(k*Pi^n*(W(2*k*Pi+1)^(2*n-1)))] 
> * [Sum_{m=0,...,n-1} (-1)^(n-1)*A042977(n-1,m)*W(2*k*Pi)^m]},
>
>if I have not made any typos. Could someone, hopefully Robert Isreal, since
>he has direct knowledge of (1) as well as his similar Maple version of (2),
>please verify (2) and, if the result is true, then perhaps add a comment or
>formula to A066318 along these lines, since I am not qualified to do so?
>
>Finally, a triangle comprising an unsigned version of A042977 might then be
>useful in OEIS, if someone would like to submit that.
>
>Thanks,
>
>LEJ
>
>REFERENCEtS:
>[1]  N. J. A. Sloane, Sequence A066318 in OEIS. http://oeis.org/A066318
>[2]  N. J. A. Sloane, Sequence A042977 in OEIS. http://oeis.org/A042977
>[3]  Robert Isreal, Answer to Challenge Problem No. 1,
>http://www.math.ubc.ca/~israel/challenge/challenge1.html
>
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