[seqfan] Re: help needed for number of 2-covers

[Robert Gerbicz]

2011/7/3 N. J. A. Sloane <njas at research.att.com>

> Dear Seq Fans, In this paper
>
> Cameron, Peter; Prellberg, Thomas; and Stark, Dudley; Asymptotic
> enumeration of 2-covers and line graphs. Discrete Math. 310 (2010), no. 2,
> 230-240
>
> there is a formula for the number v_n of restricted proper 2-covers of
> [1,..,n]:
>
> v_n = (n!/e) * Sum_{m=0..oo} (1/m!) * Sum_{k=0..n} (-1/2)^k*binomial(
> m*(m-1)/2, n-k)/k!
>
> which they get by expanding the exponential generating function
>
> V(x) = Sum_{n=0..oo} v_n x^n / n!
>
> = exp(-1-x/2) Sum_{m=0..oo} (1+x)^(m(m-1)/2) / m!
>
> They refer to Goulden and Jackson, Combinat. Enum., p. 203, as their
> source.
>
> I would like to calculate the values v_0, v_1, ... so that I can
> see which sequence it is, and also so that I can calculate several
> other sequences in their paper that are expressed in terms of v_n.
>
> But I was unable to get Maple to evaluate v_n. Can someone help?
>
> Neil
>
>
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>
That is A060053.
(used only m<1000 for small n values to get v_n for small n values).