A026945 A bisection of the Motzkin numbers A001006.

1, 2, 9, 51, 323, 2188, 15511, 113634, 853467, 6536382, 50852019, 400763223, 3192727797, 25669818476, 208023278209, 1697385471211, 13933569346707, 114988706524270, 953467954114363, 7939655757745265, 66368199913921497

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

a(n) is the sum of the squares of numbers in row n of array T given by A026300.

Number of closed walks of length 2n on the one-way infinite ladder graph starting from (and ending at) a node of degree 2. - Mitch Harris, Mar 06 2004

a(n) is the number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more noncrossing arcs. For example, with arcs separated by dashes, a(2)=9 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 14-23. - David Callan, Sep 18 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, and James Mitchell, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv preprint arXiv:1507.04838 [math.CO], 2015-2018.
• Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
• Rosa Orellana, Nancy Wallace, and Mike Zabrocki, Quasipartition and planar quasipartition algebras, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 50. See p. 7.
• Willow Stewart and Daniel Tubbenhauer, Representation gaps of rigid planar diagram monoids, arXiv:2505.05846 [math.RT], 2025. See references.
• Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

Crossrefs

Cf. A001006, A099250.

Programs

Maple

G:=(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2): GG:=series(G,x=0,60): 1, seq(coeff(GG,x^(2*n)),n=1..23);
a := n -> hypergeom([1/2-n, -n], [2], 4);
seq(simplify(a(n)), n=0..29); # Peter Luschny, May 15 2016

Mathematica

Table[SeriesCoefficient[(1-x-Sqrt[1-2*x-3*x^2])/(2*x^2),{x,0,2*n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y[0] == 1, y[1] == 1}]];
Table[MotzkinNumber[2n], {n, 0, 20}] (* Jean-François Alcover, Oct 27 2021 *)

PARI

C(n)=binomial(2*n,n)/(n+1);
a(n)=sum(k=0,n, binomial(2*n,2*k)*C(k));
\\ Joerg Arndt, May 04 2013

PARI

{a(n)=polcoeff(1/x*serreverse( x * (1-x) * (1-2*x)^2 /(1 - 3*x + 3*x^2 +x^2*O(x^n)) ),n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Oct 03 2014

Formula

a(n) = A005043(2n) + A005043(2n+1). - Ralf Stephan, Feb 06 2004

a(n) = Sum_{k=0..n} binomial(2n,2k)*C(k), C(n)=A000108(n); - Paul Barry, Jul 11 2008

a(n) = (2/Pi)*integral(x=-1..1, (1+2*x)^(2*n)*sqrt(1-x^2)). - Peter Luschny, Sep 11 2011

D-finite with recurrence: (n+1)*(2*n+1)*a(n) = (14*n^2+9*n-2)*a(n-1) + 3*(14*n^2-51*n+43)*a(n-2) - 27*(n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ 3^(2*n+3/2)/(2^(5/2)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012

G.f.: (1/x) * Series_Reversion( x * (1-x) * (1-2*x)^2 / (1 - 3*x + 3*x^2) ). - Paul D. Hanna, Oct 03 2014

From Peter Luschny, May 15 2016: (Start)

a(n) = ((9-9*n)*(2*n-3)*(4*n+1)*a(n-2)+((8*n-2))*(10*n^2-5*n-3)*a(n-1))/((1+2*n)*(4*n-3)*(n+1)) for n>=2.

a(n) = hypergeom([1/2-n, -n], [2], 4). (End)

Extensions

Entry revised by N. J. A. Sloane, Nov 16 2004