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A091964 Number of left factors of peakless Motzkin paths of length n (i.e., 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). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) is 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

REFERENCES

A. Nkwanta, Lattice paths and RNA secondary structures, DIMACS Series in Discrete Math. and Theoretical Computer Science, 34, 1997, 137-147.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, "Iterated integrals of Faulhaber polynomials and some properties of their roots", in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018.

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).

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

Conjecture: (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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A004148, A104559, A124428.

Sequence in context: A296201 A027826 A261664 * A092423 A238438 A257104

Adjacent sequences:  A091961 A091962 A091963 * A091965 A091966 A091967

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Mar 13 2004

STATUS

approved

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Last modified October 19 12:07 EDT 2018. Contains 316360 sequences. (Running on oeis4.)