[seqfan] Another possible generating function for A066318.

[Ed Jeffery]

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

REFERENCES:
[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