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A002776 - 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>A002776 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A002776" 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=%2fA002776">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="A002776 - 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> A002776 </div> <div class=seqname> Terms in certain determinants. <br><font size=-1>(Formerly M3972 N1642)</font> </div> </div> <div class=scorerefs> 2 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 5, 34, 258, 2136, 19320, 190800, 2051280, 23909760, 300827520, 4067884800, 58877280000, 908666035200, 14901260774400, 258832346572800, 4748165630208000, 91746433658880000, 1862735060938752000, 39649900359573504000, 883021783867711488000</div> <div class=seqdatalinks> (<a href="/A002776/list">list</a>; <a href="/A002776/graph">graph</a>; <a href="/search?q=A002776+-id:A002776">refs</a>; <a href="/A002776/listen">listen</a>; <a href="/history?seq=A002776">history</a>; <a href="/search?q=id:A002776&fmt=text">text</a>; <a href="/A002776/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>a(n) equals (n+1)^2 times the permanent of the (n+1) X (n+1) matrix with 1/(n+1) in the top right corner, 1/(n+1) in the bottom left corner, and 1's everywhere else. - <a href="/wiki/User:John_M._Campbell">John M. Campbell</a>, May 25 2011</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Muniru A Asiru, <a href="/A002776/b002776.txt">Table of n, a(n) for n = 0..100</a></div> <div class=sectline>J. D. H. Dickson, <a href="https://doi.org/10.1112/plms/s1-10.1.120">Discussion of two double series arising from the number of terms in determinants of certain forms</a>, Proc. London Math. Soc., 10 (1879), 120-122.</div> <div class=sectline>J. D. H. Dickson, <a href="/A002775/a002775.pdf">Discussion of two double series arising from the number of terms in determinants of certain forms</a>, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (n^3 + n^2 + 2*n + 1)*n!.</div> <div class=sectline>a(n) = (n+3)! - 5*(n+2)! + 6*(n+1)! - n!. - <a href="/wiki/User:Umut_脰zer">Umut 脰zer</a>, Dec 26 2017</div> <div class=sectline>E.g.f.: (1 + x + 3*x^2 + x^3)/(1 - x)^4. - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Apr 17 2022</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A002776" title="Terms in certain determinants.">A002776</a> := [seq(factorial(n+3) - 5 * factorial(n+2) + 6 * factorial(n+1) - factorial(n), n=0..100)]; # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Jan 15 2018</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[(n^3+n^2+2n+1)n!, {n, 0, 30}] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Oct 28 2011 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(GAP) <a href="/A002776" title="Terms in certain determinants.">A002776</a> := List([0..100], n -> Factorial(n+3) - 5 * Factorial(n+2) + 6 * Factorial(n+1) - Factorial(n)); # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Jan 15 2018</div> <div class=sectline>(Magma) [(n^3+n^2+2*n+1)*Factorial(n): n in [0..20]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jan 19 2018</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A047922" title="Triangle of numbers a(n,k) = number of terms in n X n determinant with 2 adjacent diagonals of k and k-1 0's (0<=k<=n).">A047922</a>.</div> <div class=sectline>Sequence in context: <a href="/A080503" title="a(n) = A080502(n)/n.">A080503</a> <a href="/A078284" title="a(n) = A078283(n)/n.">A078284</a> <a href="/A083987" title="a(n) = n-th partial concatenation of A051883 divided by n.">A083987</a> * <a href="/A333804" title="(1/16) times the number of permutations p of [n+2] such that |p(i+n) - p(i)| <> n for i in {1,2}.">A333804</a> <a href="/A081342" title="a(n) = (8^n + 2^n)/2.">A081342</a> <a href="/A248055" title="G.f.: Sum_{n>=0} x^n / (1-4*x)^(2*n+1) * [Sum_{k=0..n} C(n,k)^2 * x^k] * [Sum_{k=0..n} C(n,k)^2 * 4^k * x^k].">A248055</a></div> <div class=sectline>Adjacent sequences: <a href="/A002773" title="Number of nonisomorphic simple matroids (or geometries) with n points.">A002773</a> <a href="/A002774" title="Number of bipartite partitions of n white objects and n black ones.">A002774</a> <a href="/A002775" title="a(n) = n^2 * n!.">A002775</a> * <a href="/A002777" title="Restricted permutations.">A002777</a> <a href="/A002778" title="Numbers whose square is a palindrome.">A002778</a> <a href="/A002779" title="Palindromic squares.">A002779</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>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Edited by <a href="/wiki/User:Dean_Hickerson">Dean Hickerson</a>, Sep 20 2002</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 April 10 05:25 EDT 2025. Contains 382654 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>