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A008584 - 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>A008584 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A008584" 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=%2fA008584">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="A008584 - 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> A008584 </div> <div class=seqname> Molien series for Weyl group E_6. </div> </div> <div class=scorerefs> 4 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 6, 4, 8, 6, 10, 9, 14, 11, 18, 15, 22, 20, 29, 25, 36, 32, 43, 41, 54, 49, 66, 61, 78, 75, 95, 89, 113, 107, 132, 129, 157, 150, 184, 178, 212, 209, 248, 241, 287, 280, 327</div> <div class=seqdatalinks> (<a href="/A008584/list">list</a>; <a href="/A008584/graph">graph</a>; <a href="/search?q=A008584+-id:A008584">refs</a>; <a href="/A008584/listen">listen</a>; <a href="/history?seq=A008584">history</a>; <a href="/search?q=id:A008584&fmt=text">text</a>; <a href="/A008584/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,7</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.</div> <div class=sectline>H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups. Ergebnisse der Mathematik und Ihrer Grenzgebiete, New Series, no.14. Springer Verlag, 1957, Table 10.</div> <div class=sectline>L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 35).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A008584/b008584.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=248">Encyclopedia of Combinatorial Structures 248</a></div> <div class=sectline><a href="/index/Mo#Molien">Index entries for Molien series</a></div> <div class=sectline><a href="/index/Rec#order_42">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,0,1,1,-1,0,1,-1,-2,1,0,-3,0,2,-1,-1,3,1,-2,1,3,-1,-1,2,0,-3,0,1,-2,-1,1,0,-1,1,1,0,0,1,0,-1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: 1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)).</div> <div class=sectline>a(n) ~ 1/6220800*n^5 + 1/414720*n^4. - <a href="/wiki/User:Ralf_Stephan">Ralf Stephan</a>, Apr 29 2014</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>seq(coeff(series(1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)), x, n+1), x, n), n = 0..60); # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Jan 31 2020</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>CoefficientList[Series[1/((1-x^2)(1-x^5)(1-x^6)(1-x^8)(1-x^9)(1-x^12)), {x, 0, 55}], x] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Aug 10 2011 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Magma) MolienSeries(CoxeterGroup("E6")); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006</div> <div class=sectline>(PARI) my(x='x+O('x^60)); Vec(1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12))) \\ <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Jan 31 2020</div> <div class=sectline>(Sage)</div> <div class=sectline>def <a href="/A008584" title="Molien series for Weyl group E_6.">A008584</a>_list(prec):</div> <div class=sectline> P.<x> = PowerSeriesRing(ZZ, prec)</div> <div class=sectline> return P( 1/((1-x^2)*(1-x^5)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)) ).list()</div> <div class=sectline><a href="/A008584" title="Molien series for Weyl group E_6.">A008584</a>_list(60) # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Jan 31 2020</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A014977" title="Expansion of Molien series for automorphism group (2.Weyl(E6)) of E6 lattice.">A014977</a>.</div> <div class=sectline>Sequence in context: <a href="/A115584" title="Number of partitions of n in which each part k occurs more than k times.">A115584</a> <a href="/A058742" title="McKay-Thompson series of class 68A for Monster.">A058742</a> <a href="/A029140" title="Expansion of 1/((1-x^2)(1-x^3)(1-x^4)(1-x^10)).">A029140</a> * <a href="/A352833" title="Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k fixed points, k = 0, 1.">A352833</a> <a href="/A034390" title="Multiplicity of highest weight (or singular) vectors associated with character chi_2 of Monster module.">A034390</a> <a href="/A368671" title="For any k >= 0, let P(k) = A368357(k) and P(-k) = A368358(k); for any n > 0, a(n) is the unique k such that P(k) = n.">A368671</a></div> <div class=sectline>Adjacent sequences: <a href="/A008581" title="Molien series for 6-dimensional complex reflection group 4.U_4 (3) of order 2^9 .3^7 .5.7.">A008581</a> <a href="/A008582" title="Molien series for Weyl group E_8.">A008582</a> <a href="/A008583" title="Molien series for Weyl group E_7.">A008583</a> * <a href="/A008585" title="a(n) = 3*n.">A008585</a> <a href="/A008586" title="Multiples of 4.">A008586</a> <a href="/A008587" title="Multiples of 5: a(n) = 5 * n.">A008587</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="it is very easy to produce terms of sequence">easy</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:N._J._A._Sloane">N. J. A. Sloane</a></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 30 20:37 EST 2024. Contains 378265 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>