A091964 Number of left factors of peakless Motzkin paths of length n.

1, 2, 4, 9, 21, 50, 121, 296, 730, 1812, 4521, 11328, 28485, 71844, 181674, 460443, 1169283, 2974574, 7578937, 19337489, 49401526, 126350742, 323495259, 829033334, 2126454271, 5458711430, 14023219126, 36049991901, 92734505565

Offset: 0

Author

Emeric Deutsch, Mar 13 2004

Keywords

• nonn

Comments

Number of paths from (0,0) to the line x=n, consisting of steps u=(1,1), h=(1,0), d=(1,-1), that never go below the x-axis and a u step is never followed by a d step.

a(n) is also the number of peakless Motzkin paths of length n in which the (1,0)-steps at level 0 come in 2 colors. Example: a(4)=21 because, denoting u=(1,1), h=(1,0), and d=(1,-1), we have 2^4 = 16 paths of shape hhhh, 2 paths of shape huhd, 2 paths of shape uhdh, and 1 path of shape uhhd. - Emeric Deutsch, May 03 2011

Equals diagonal sums of triangle A124428. - Paul D. Hanna, Oct 31 2006

Examples

			a(2)=4 because we have hh, hu, uh and uu.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.
• Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
• Ivo L. Hofacker, Christian M. Reidys, and Peter F. Stadler, Symmetric circular matchings and RNA folding. Discr. Math., 312:100-112, 2012. See Prop. 5, C_2^{1}(z).
• Asamoah Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.
• Helmut Prodinger, Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs, arXiv:2501.13645 [math.CO], 2025. See p. 8.

Crossrefs

Cf. A004148, A104559, A124428.

Programs

Magma

[(&+[Binomial(Floor((n+k)/2),k)*Binomial(Floor((n+k+1)/2),k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 26 2019

Mathematica

CoefficientList[Series[2/(1-3*x+x^2+Sqrt[1-2*x-x^2-2*x^3+x^4]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

PARI

a(n)=sum(k=0,n,binomial(n-k\2,(k+1)\2)*binomial(n-(k+1)\2,k\2)) \\ Paul D. Hanna, Mar 24 2005

PARI

a(n)=sum(k=0,n,binomial((n+k)\2,k)*binomial((n+k+1)\2,k)) \\ Paul D. Hanna, Oct 31 2006

Sage

[sum(binomial(floor((n+k)/2),k)*binomial(floor((n+k+1)/2),k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Feb 26 2019

Formula

G.f.: 2/(1 - 3*z + z^2 + sqrt(1 - 2*z - z^2 - 2*z^3 + z^4)).

a(n) = Sum_{k=0..n} C(n-floor(k/2), floor((k+1)/2)) * C(n-floor((k+1)/2), floor(k/2)). - Paul D. Hanna, Mar 24 2005

a(n) = Sum_{k=0..n} C(floor((n+k)/2),k)*C(floor((n+k+1)/2),k). - Paul D. Hanna, Oct 31 2006

G.f.: 1/(1-x-x/(1-x^2/(1-x/(1-x^2/(1-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Jun 30 2009

D-finite with recurrence (n+1)*a(n) + 2*(-n-1)*a(n-1) + (-n+1)*a(n-2) + 2*(-n+3)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 24 2012

a(n) ~ (3+sqrt(5))^n / (sqrt(7*sqrt(5)-15) * sqrt(Pi*n) * 2^(n-1/2)). - Vaclav Kotesovec, Feb 12 2014

Equivalently, a(n) ~ phi^(2*n + 2) / (5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021