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A259212 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A259212 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A259212" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA259212">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A259212 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A259212 </div> <div class=seqname> A total of n married couples, including a mathematician M and his wife W, are to be seated at the 2n chairs around a circular table. M and W are the first couple allowed to choose chairs, and they choose two chairs next to each other. The sequence gives the number of ways of seating the remaining couples so that women and men are in alternate chairs but M and W are the only couple seated next to each other. </div> </div> <div class=scorerefs> 8 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 0, 0, 6, 72, 1920, 69120, 3402000, 218252160, 17708544000, 1773002649600, 214725759494400, 30941575378560000, 5231894853375590400, 1025881591718766182400, 230901375630648602880000, 59127083048250564931584000, 17091634972762948947148800000</div> <div class=seqdatalinks> (<a href="/A259212/list">list</a>; <a href="/A259212/graph">graph</a>; <a href="/search?q=A259212+-id:A259212">refs</a>; <a href="/A259212/listen">listen</a>; <a href="/history?seq=A259212">history</a>; <a href="/search?q=id:A259212&fmt=text">text</a>; <a href="/A259212/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,4</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>After M and W are seated at neighboring chairs, the problem of enumerating the ways of seating the remaining n-1 married couples is equivalent to the following problem: find the number of ways of seating n-1 married couples at 2*(n-1) chairs in a straight line, men and women in alternate chairs, so that no husband is next to his wife. According to our comment in <a href="/A000271" title="Sums of m茅nage numbers.">A000271</a>, this problem has a solution 2*(n-1)!*<a href="/A000271" title="Sums of m茅nage numbers.">A000271</a>(n-1), n >= 2. Here the coefficient 2 should be replaced by 1, since the place of the first woman W, by the condition, is already fixed.</div> <div class=sectline>Also the number of Hamiltonian paths in the (n-1)-crown graph for n > 3. - <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Mar 27 2018</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A259212/b259212.txt">Table of n, a(n) for n=1..18.</a></div> <div class=sectline>Vladimir Shevelev and Peter J. C. Moses, <a href="http://www.emis.de/journals/INTEGERS/papers/q72/q72.Abstract.html">Alice and Bob go to dinner: A variation on menage</a>, INTEGERS, Vol. 16(2016), #A72.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>.</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (n-1)!*<a href="/A000271" title="Sums of m茅nage numbers.">A000271</a>(n-1), for n > 1.</div> <div class=sectline>From <a href="/wiki/User:Vladimir_Shevelev">Vladimir Shevelev</a>, Jul 07 2015: (Start)</div> <div class=sectline>Consider the general formula for solution A(r,n) in <a href="/A258673" title="A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular ...">A258673</a> without the restriction A(r,n)=0 for n <= (d+1)/2 in case d=2*n-1. The case when M and W sit at neighboring chairs corresponds to d=1, r=2 or d=2*n-1, r=n+1. In both cases, from this formula we have</div> <div class=sectline>A(r,n) = a(n)/(n-1)! = Sum_{j=0..n-1} (-1)^j * binomial(2*n-j-2,j)*(n-j-1)!, n > 1. (End)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>a[n_] := (n-1)! Sum[(-1)^(n-k+1) k! Binomial[n+k-1, 2k], {k, 0, n}]; a[1] = 0; Array[a, 18] (* <a href="/wiki/User:Jean-Fran莽ois_Alcover">Jean-Fran莽ois Alcover</a>, Sep 03 2016 *)</div> <div class=sectline>Join[{0}, Table[-(-1)^n (n - 1)! HypergeometricPFQ[{1, 1 - n, n}, {1/2}, 1/4], {n, 2, 20}]] (* <a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a>, Mar 27 2018 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n) = if (n==1, 0, my(m=n-1); m!*sum(k=0, m, binomial(2*m-k, k)*(m-k)!*(-1)^k)); \\ <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, Jun 26 2015</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000179" title="M茅nage numbers: a(0) = 1, a(1) = -1, and for n >= 2, a(n) = number of permutations s of [0, ..., n-1] such that s(i) != i a...">A000179</a>, <a href="/A000271" title="Sums of m茅nage numbers.">A000271</a>, <a href="/A258664" title="A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular ...">A258664</a>, <a href="/A258665" title="A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular ...">A258665</a>, <a href="/A258666" title="A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular ...">A258666</a>, <a href="/A258667" title="A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular ...">A258667</a>, <a href="/A258673" title="A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular ...">A258673</a>.</div> <div class=sectline>Sequence in context: <a href="/A120331" title="Number of polynomials in two noncommuting variables over the field F_2 having complexity or rank n.">A120331</a> <a href="/A009523" title="Expansion of sin(tanh(x))*cos(x).">A009523</a> <a href="/A009793" title="Expansion of tanh(sin(x))*cos(x).">A009793</a> * <a href="/A279234" title="Denominators of coefficients in expansion of 3/(2 + cos(sqrt(x))).">A279234</a> <a href="/A132878" title="Column 1 of triangle A132875.">A132878</a> <a href="/A006585" title="Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n.">A006585</a></div> <div class=sectline>Adjacent sequences: <a href="/A259209" title="E.g.f.: Sum_{n>=1} x^(n^2) * exp(x^n) / n!.">A259209</a> <a href="/A259210" title="Positive hexagonal numbers (A000384) that are other hexagonal numbers divided by 3.">A259210</a> <a href="/A259211" title="Positive octagonal numbers (A000567) that are other octagonal numbers divided by 3.">A259211</a> * <a href="/A259213" title="Column 2 of triangle A258829.">A259213</a> <a href="/A259214" title="Number of (n+1)X(n+1) 0..1 arrays with each 2X2 subblock having clockwise pattern 0000 0011 or 0101">A259214</a> <a href="/A259215" title="Number of (n+1) X (1+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.">A259215</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="an exceptionally nice sequence">nice</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Vladimir_Shevelev">Vladimir Shevelev</a>, Jun 21 2015</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>More terms from <a href="/wiki/User:Peter_J._C._Moses">Peter J. C. Moses</a>, Jun 21 2015</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified November 24 12:50 EST 2024. Contains 378082 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>