update to A094047: arrangements of n couples

Max Alekseyev maxale at gmail.com
Mon Feb 11 08:34:54 CET 2008

```Neil,

%I A137801
%S A137801 0,6,900,748440,1559930400,6928346502000,58160619655538400,845986566719614320000,
%T A137801 19957466912796971445888000,724891264860942581350908960000,38873628093261330554954970801600000
%U A137801 2973438561847648316247471320092978560000,315157381000427102303851602112455099840000000
%N A137801 Number of arrangements of 2n couples into n cars such that
each car contains 2 men and 2 women but no couple (cars are labeled).
%O A137801 1,2
%F A137801 a(n) = n! * A137802(n) = n! * SUM[i+j<=n] (-1)^i * (2n)! *
(2n-i-2j)! / (n-i-j)! / i! / j! / 2^(2n-2i-j)
%H A137801 <a href="http://lib.mexmat.ru/forum/viewtopic.php?p=92408#92408">Proof
of the formula</a> (in Russian).
%o A137801 (PARI) { a(n) = n! * sum(i=0,n, (-1)^i * sum(j=0,n-i,
(2*n)! * (2*n-i-2*j)! / (n-i-j)! / i! / j! / 2^(2*n-2*i-j) ) ) }
%Y A137801 Cf. A094047, A137802.
%K A137801 nonn
%A A137801 Max Alekseyev (maxal(AT)cs.ucsd.edu), Feb 10 2008

%I A137802
%S A137802 0,3,150,31185,12999420,9622703475,11539805487210,20981809690466625,54997428661808232600,
%T A137802 199760599884519009411075,973866344327734952575230750,6207575427404936259602204502225,
%U A137802 50611261969837502759562261637612500,518063740946674946003070940552409581875
%N A137802 Number of arrangements of 2n couples into n cars such that
each car contains 2 men and 2 women but no couple (cars are
unlabeled).
%O A137802 1,2
%F A137802 a(n) = A137801(n) / n! = SUM[i+j<=n] (-1)^i * (2n)! *
(2n-i-2j)! / (n-i-j)! / i! / j! / 2^(2n-2i-j)
%o A137802 (PARI) { a(n) = sum(i=0,n, (-1)^i * sum(j=0,n-i, (2*n)! *
(2*n-i-2*j)! / (n-i-j)! / i! / j! / 2^(2*n-2*i-j) ) ) }
%Y A137802 Cf. A094047, A137801.
%K A137802 nonn
%A A137802 Max Alekseyev (maxal(AT)cs.ucsd.edu), Feb 10 2008

Regards,
Max

On Feb 10, 2008 10:09 PM, Max Alekseyev <maxale at gmail.com> wrote:
> Neil, SeqFans,
>
> I propose the following update to A094047 based on the formula that
> I've got for its terms.
>
> Regards,
> Max
>
>
> %I A094047
> %S A094047 0,0,2,12,312,9600,416880,23879520,1749363840,159591720960,17747520940800,2363738855385600,
> %T A094047 371511874881100800,68045361697964851200,14367543450324474009600,3464541314885011705344000,
> %U A094047 946263209467217020194816000,290616691739323132839591936000,99714530592486301158487941120000
> %N A094047 Number of arrangements of n couples around a round table so
> that each person sits between two people of the opposite gender and no
> couple is seated together.
> %F A094047 For n>1, a(n) = (-1)^n * 2 * (n-1)! + n! * SUM[j=0..n-1]
> (-1)^j * (n-j-1)! * binomial(2*n-j-1,j).
> %C A094047 Also, the number of Hamiltonian directed circuits in the
> crown graph of order n.
> %H A094047 Eric Weisstein's World of Mathematics, <a
> href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>
> %H A094047 Eric Weisstein's World of Mathematics, <a
> href="http://mathworld.wolfram.com/HamiltonianCircuit.html">Hamiltonian
> Circuit</a>
> %Y A094047 Cf. A114939, A137729.
> %Y A094047 Adjacent sequences: A094044 A094045 A094046 this_sequence
> A094048 A094049 A094050
> %Y A094047 Sequence in context: A012425 A012422 A122767 this_sequence
> A091472 A012727 A088229
> %K A094047 nonn
> %O A094047 1,3
> %A A094047 Matthijs Coster (matthijs(AT)coster.demon.nl), Apr 29 2004
> %E A094047 Better definition from Joel Lewis
> (jblewis(AT)post.harvard.edu), Jun 30 2007
> %E A094047 Formula and further terms from Max Alekseyev
> (maxal(AT)cs.ucsd.edu), Feb 10 2008
>

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