A135045: missed terms(?)

[Richard Mathar]

something very different must have been in the mind of the submitter.
  degenerate, that is, 2 times a single term of A001043 (doesn't match)
  also be sums of three terms (doesn't match).
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Date: Fri, 15 Feb 2008 10:24:08 -0800
From: "Max Alekseyev" <maxale at gmail.com>
To: SeqFan <seqfan at ext.jussieu.fr>
Subject: update to A114939: yet another seating arrangements for n couples
Cc: "N. J. A. Sloane" <njas at research.att.com>,
   "Hugo Pfoertner" <all at abouthugo.de>
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Dear Neil, Hugo, SeqFans,

This is an updated entry A114939 based on the formula that I've got
for this sequence.
The formula is rather complicated (I'm still thinking on how to
simplify it), so I present it only in the form of PARI/GP program.
It turned out that the listed valued of A114939 for n>=4 were
incorrect, that was acknowledged by Hugo, who has also confirmed the
terms a(4..7) computed by my formula.

Neil, this is the revised entry.

%I A114939
%S A114939 0,1,7,216,10956,803400,83003040,11579823360,2080493573760,469031859192960,
%U A114939 129727461014726400,43176116371928601600,17025803126147196057600,
%T A114939 7850538273249476117913600,4184985289634680301509939200
%N A114939 Number of essentially different seating arrangements for n
couples around a circular table with 2*n seats avoiding spouses being
neighbors and avoiding clusters of 3 persons with equal gender.
%C A114939 Arrangements that differ only by rotation or reflection are
excluded by the following conditions: Seat number 1 is assigned to
person (a). Person (a)'s spouse (A) can only take seats with numbers
<=(n+1). If (A) gets seat n+1 (i.e. sits exactly opposite to her/his
spouse) then person (B) can only take seats with numbers <= n.
%F A114939 See PARI code for the formula.
%e A114939 a(2)=1 because the only valid arrangement is aBAb. a(3)=7
because the only valid arrangements under the given conditions are:
abAcBC, aBAcbC, aBcAbC, aBcACb, acAbCB, acBAbC, aCAbcB.
%o A114939 (PARI) {  a(n) = if(n<=1, 0, (-1)^n*(n-1)!*2^(n-1) + n! *
polcoeff( polcoeff( [0, 2*y*z^3 + z^2, -3*y*z^5 - 4*z^4 + ((-2*y^2 -
1)/y)*z^3, 6*y*z^7 + (4*y^2 + 11)*z^6 + ((8*y^2 + 4)/y)*z^5 + 3*z^4] *
sum(j=0,n-1, j! * [0, 0, 0, -z^6 + z^4; 1, 0, 0, ((y^2 + 1)/y)*z^5 -
2*z^4 + ((-y^2 - 1)/y)*z^3; 0, 1, 0, ((2*y^2 + 2)/y)*z^3 + z^2; 0, 0,
1, -2*z^2]^(n+j) ) * [1,0,0,0]~, 2*n,z), 0,y) / 2 ); }
%Y A114939 Cf. A114938, A137729, A137730, A137737, A137749
%Y A114939 Adjacent sequences: A114936 A114937 A114938 this_sequence
A114940 A114941 A114942
%Y A114939 Sequence in context: A120661 A086214 A133589 this_sequence
A119942 A130741 A003385
%K A114939 nonn
%O A114939 1,3
%A A114939 Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 08 2006
%E A114939 a(4..7) corrected, formula and further term provided by Max
Alekseyev (maxal(AT)cs.ucsd.edu), Feb 15 2008

Regards,
Max