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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

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 February 23 11:32 EST 2018. Contains 299579 sequences. (Running on oeis4.)