[A000179]

[David Wilson]

By computing from the definition I get

1, 0, 0, 1, 2, 13, 80, ...

a(0) = 1; () works.

a(1) = 0; nothing works.

a(2) = 0; nothing works.

a(3) = 1; (201) works.

a(4) = 2; (2301), (3012) work.

a(5) = 13; (20413), (23401), (24013), (24103), (30412), (30421), (34012),
   (34021), (34102), (40123), (43012), (43021), (43102) work.

I contend A000179 should start (1, 0, 0, 1, 2, 13, 80, ...)

The recurrence works only for n >= 4.  To make it work for all n >= 2, we
must
start the sequence (2, -1, 0, 1, 2, 13, 80, ...).  It doesn't seem right
that a
combinatorial sequence should include a negative number.

I also contend that A059375 should start (1, 0, 0, 12, 96, 3120, 115200,
...),
that is, a(0) = 1, a(1) = 0.  This properly counts solutions to the menage
problem.  Then we have A059375(n) = 2*n!*A000197(n) for all n >= 1.

----- Original Message -----
From: "Christian G. Bower" <bowerc at usa.net>
To: "seqfan" <seqfan at ext.jussieu.fr>
Sent: Friday, February 07, 2003 9:26 PM
Subject: Re: [A000179]


"David Wilson" <davidwwilson at attbi.com> wrote:
> ---------------------------------------------
> Attachment:
> MIME Type: multipart/alternative
> ---------------------------------------------
> In the OEIS, A000179 starts out 1, 1, 0, 1, 2, 13, 80, ...
>
> In the Mathworld article "Married Couples Problem", A000179 starts out -1,
1, 0, 2, 13, 80, ...
>
> Which is correct?
>

Reading the description in the EIS, I would think a_4 = 1, the only
qualifying permutation being 2301.

a_5 should be 2 with 23401 and 34012

before a_4 there cannot be any. From the Mathworld description, it looks
like terms before a_4 may have been assigned to make the formulas or
recurrences work. I'm having a hard time getting reasonable values out
of any of the formulas. (Maybe I'm making elementary errors.)

In any case, it appears the EIS starting index should be 1, not 0. After
the 0 at a_3, it should be 1, 2, ... as in the EIS, not 2, ... as in
Mathworld and if anybody can figure out whether the terms before the
zero are better as 1,1 or -1,1 or 0,0 or something else, please speak up.

Christian