[seqfan] Re: Reproducing A002771 skew determinants

[Sean A. Irvine]

To answer my own question, have now found the formula to generate
A002771 after looking at A002772 (it is given on pg. 278 of vol 4 of
Muir:

s:=n->sum(binomial(n,2*k)*doublefactorial(2*k-1)*(1+doublefactorial(2*k-1))/2,k=0..floor(n/2));

http://quod.lib.umich.edu/cache/a/c/m/acm9350.0004.001/00000301.tif.20.pdf#page=13;zoom=75

Still find it interesting that the sequence differs from the other
formulas, but I don't know enough about what is being counted to
determine which is "right" in that sense.

I'll update the entry accordingly.

Sean.

On Tue, Aug 19, 2014 at 9:03 AM, Sean A. Irvine <sairvin at xtra.co.nz> wrote:
> I have been trying to reproduce the terms of A002771 (terms in a skew
> determinant).
>
> Based on Sylvester (1879), an exponential generating function for the
> sequence is exp(x+x^2/4) / (1-x^2)^(1/4).
>
> Using the example in Muir (1906-1923), another formula appears to be
>
> a(n) = sum(binomial(n, n - 2*k) * A002370(k), k=0..floor(n/2)).
>
> Both the egf and summation formula produce the following sequence:
>
> 1, 2, 4, 13, 41, 226, 1072, 9059, 58123, 657766, 5268836, 73980787, ...
>
> but this disagrees with the OEIS after term 7:
>
> 1, 2, 4, 13, 41, 226, 1072, 9374, 60958, 723916, 5892536, 86402812, ...
>
> Page 282 of Muir only lists the sequence up to 1072:
>
> http://quod.lib.umich.edu/cache/a/c/m/acm9350.0003.001/00000301.tif.20.pdf#page=11;zoom=75
>
> So my question is where did the current terms in the OEIS after 1072 come
> from?  Does anyone have access to Muir (1896) which might be the original
> source?
>
> I am loathe to change the terms of such as old sequence.  Would the correct
> course of action be to create a new sequence using above formulas and link
> the original as a potential erroneous version?
>
> Finally, the Rivera (2014) paper attached to A002771 does not appear to be
> relevant and I could see no reference to the sequence in that paper.  It
> does mention, A027771 -- so perhaps it has been attached to the wrong
> sequence?
>
> Regards,
> Sean.
>
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