[seqfan] Re: A002831 incompatible with A002830

[Max Alekseyev]

Btw, this is a type of Euler transform -- the third one described at
http://mathworld.wolfram.com/EulerTransform.html
Max


On Tue, Sep 9, 2014 at 2:06 AM, Sean A. Irvine <sairvin at xtra.co.nz> wrote:
> The terms of A002831 do not match the description:
>
> A002831:  1, 4, 11, 60, 362, 2987
> Computed: 1, 4, 11, 60, 318, 2806, 23445, 314518
>
> If F(x) is the generating function for A002831, then we have
>
> G(x) = exp(sum(F(x^k)/k, k=1..infinity))
>
> is the generating function for A002830.  That is, this transform
> permits going from a count of connected graphs to the corresponding
> count of total graphs (i.e. included disconnected cases). [As an
> aside the corresponding inverse is by Mobius inversion
> F(x) = sum(mu(k) * log(G(x^k)) / k, k=1..infinity).]
>
> Applying the g.f. transformation to the "Computed" values yields the
> current values for A002830 giving me confidence that the computed
> values are correct.
>
> I don't have access to R. C. Read's dissertation, so I cannot verify
> where the existing values for a(5) and a(6) came from.
>
> Should I update the entry for A002831 (which some comment about the
> original terms), or is some other action appropriate?
>
> Regards,
> Sean A. Irvine
>
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