[seqfan] help needed for number of 2-covers

N. J. A. Sloane njas at research.att.com
Sun Jul 3 16:42:44 CEST 2011

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?


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