CINXE.COM
A014531 - 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>A014531 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A014531" 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=%2fA014531">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="A014531 - 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> A014531 </div> <div class=seqname> Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 2nd column from the center. </div> </div> <div class=scorerefs> 15 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 3, 10, 30, 90, 266, 784, 2304, 6765, 19855, 58278, 171106, 502593, 1477035, 4343160, 12778152, 37616427, 110797569, 326527350, 962803170, 2840372304, 8383467708, 24755608584, 73133433800, 216143407675, 639062383401</div> <div class=seqdatalinks> (<a href="/A014531/list">list</a>; <a href="/A014531/graph">graph</a>; <a href="/search?q=A014531+-id:A014531">refs</a>; <a href="/A014531/listen">listen</a>; <a href="/history?seq=A014531">history</a>; <a href="/search?q=id:A014531&fmt=text">text</a>; <a href="/A014531/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Number of "up" steps in all Motzkin paths of length n+1. E.g. a(2)=3 because in the four Motzkin paths of length 3, HHH, HUD, UDH and UHD, where H=(1,0), U=(1,1), D=(1,-1), we have altogether three U steps. - <a href="/wiki/User:Emeric_Deutsch">Emeric Deutsch</a>, Dec 26 2003</div> <div class=sectline>a(n-1) = <a href="/A111808" title="Left half of trinomial triangle (A027907), triangle read by rows.">A111808</a>(n,n-2) for n>1. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Aug 17 2005</div> <div class=sectline>a(n) = number of paths in the half-plane x>=0, from (0,0) to (n+1,2), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=2, we have the 3 paths: UUH, HUU, UHU. - <a href="/wiki/User:Jos茅_Luis_Ram铆rez_Ram铆rez">Jos茅 Luis Ram铆rez Ram铆rez</a>, Apr 19 2015</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>G. C. Greubel, <a href="/A014531/b014531.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..200 from T. D. Noe)</div> <div class=sectline>Ricardo G贸mez A铆za, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See pp. 21-22.</div> <div class=sectline>Mark Shattuck, <a href="https://doi.org/10.54550/ECA2024V4S4R32">Subword Patterns in Smooth Words</a>, Enum. Comb. Appl. (2024) Vol. 4, No. 4, Art. No. S2R32. See p. 6.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient.</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = <a href="/A002426" title="Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.">A002426</a>(n+1)-<a href="/A001006" title="Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.">A001006</a>(n+1) = a(n-1)+<a href="/A005717" title="Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.">A005717</a>(n)+<a href="/A014532" title="Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 3rd column from the center.">A014532</a>(n-2) - <a href="/wiki/User:Henry_Bottomley">Henry Bottomley</a>, May 15 2001</div> <div class=sectline>E.g.f.: exp(x)*(2*x*BesselI(1, 2*x)+(x-2)*BesselI(2, 2*x))/x. - <a href="/wiki/User:Vladeta_Jovovic">Vladeta Jovovic</a>, Aug 21 2003</div> <div class=sectline>G.f.: [1-2z-z^2-(1-z)q]/(2z^3q), where q=sqrt(1-2z-3z^2). - <a href="/wiki/User:Emeric_Deutsch">Emeric Deutsch</a>, Dec 26 2003</div> <div class=sectline>a(n) = Sum_{k=0..n+1} C(n+1,k)*C(n-k+1,k+2). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Sep 20 2004</div> <div class=sectline>D-finite with recurrence (n+3)*(n-1)*a(n) -(n+1)*(2n+1)*a(n-2)-3*n*(n+1)*a(n-2)=0. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Dec 08 2011</div> <div class=sectline>a(n) = n*(n+1)*hypergeom([(1-n)/2, 1-n/2], [3], 4)/2. - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Nov 23 2014</div> <div class=sectline>G.f.: z*M(z)^2/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - <a href="/wiki/User:Jos茅_Luis_Ram铆rez_Ram铆rez">Jos茅 Luis Ram铆rez Ram铆rez</a>, Apr 19 2015</div> <div class=sectline>a(n) = GegenbauerC(n-1, -n-1, -1/2). - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, May 09 2016</div> <div class=sectline>a(n) = Sum_{k>0} k * <a href="/A055151" title="Triangular array of Motzkin polynomial coefficients.">A055151</a>(n+1,k). - <a href="/wiki/User:Alois_P._Heinz">Alois P. Heinz</a>, Mar 29 2020</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>seq( add(binomial(i+1, k)*binomial(i-k+1, k+2), k=0..floor(i/2)), i=1..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001</div> <div class=sectline>a := n -> simplify(GegenbauerC(n-1, -n-1, -1/2)):</div> <div class=sectline>seq(a(n), n=1..26); # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, May 09 2016</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[Sum[Binomial[i + 1, k]*Binomial[i - k + 1, k + 2], {k, 0, Floor[i/2]}], {i, 30}] (* <a href="/wiki/User:Michael_De_Vlieger">Michael De Vlieger</a>, Apr 20 2015 *)</div> <div class=sectline>Table[GegenbauerC[n - 1, -n - 1, -1/2], {n, 1, 50}] (* <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Feb 28 2017 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Sage)</div> <div class=sectline>a = lambda n: n*(n+1)*hypergeometric([(1-n)/2, 1-n/2], [3], 4)/2</div> <div class=sectline>[simplify(a(n)) for n in (1..26)] # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Nov 23 2014</div> <div class=sectline>(PARI) for(n=1, 25, print1(sum(k=0, n+1, binomial(n+1, k)*binomial(n-k+1, k+2)), ", ")) \\ <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Feb 28 2017</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A027907" title="Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + ...">A027907</a>, <a href="/A005717" title="Construct triangle in which n-th row is obtained by expanding (1 + x + x^2)^n and take the next-to-central column.">A005717</a>, <a href="/A055151" title="Triangular array of Motzkin polynomial coefficients.">A055151</a>.</div> <div class=sectline>First differences are in <a href="/A025180" title="a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2...">A025180</a>.</div> <div class=sectline>Sequence in context: <a href="/A261336" title="Number of achiral molecules containing two linear or branched chiral alkyl chains of n carbon atoms each.">A261336</a> <a href="/A026109" title="a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, s(n) = 3, |s(i) ...">A026109</a> <a href="/A026327" title="a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n,...">A026327</a> * <a href="/A062107" title="Diagonal of table A062104.">A062107</a> <a href="/A269800" title="Convolution of A000107 and A027852.">A269800</a> <a href="/A033113" title="Base-3 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.">A033113</a></div> <div class=sectline>Adjacent sequences: <a href="/A014528" title="Neither == 0 (mod 4) nor a power of 3.">A014528</a> <a href="/A014529" title="Largest convex area that can be tiled with n equilateral triangles whose sides s_k are relatively prime, i.e., gcd(s_1,...,s...">A014529</a> <a href="/A014530" title="List of sizes of squares occurring in lowest order example of a perfect squared square.">A014530</a> * <a href="/A014532" title="Form array in which n-th row is obtained by expanding (1+x+x^2)^n and taking the 3rd column from the center.">A014532</a> <a href="/A014533" title="Form array in which n-th row is obtained by expanding (1 + x + x^2)^n and taking the 4th column from the center.">A014533</a> <a href="/A014534" title="Inverse of 525th cyclotomic polynomial.">A014534</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></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>More terms from <a href="/wiki/User:James_A._Sellers">James A. Sellers</a>, Feb 05 2000</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 7 11:15 EDT 2025. Contains 382563 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>