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A089372 Number of Motzkin paths of length n with no peaks at level 1. 7
1, 1, 1, 2, 5, 12, 29, 72, 183, 473, 1239, 3282, 8777, 23665, 64261, 175584, 482395, 1331795, 3692891, 10280190, 28719659, 80493514, 226268539, 637767720, 1802113489, 5103874135, 14485789561, 41194844114, 117366166381 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

Qiang-Hui Guo, LH Sun, J Wang, Regular Simple Queues of Protein Contact Maps, Bulletin of Mathematical Biology, 2016, January 2017, Volume 79, Issue 1, pp 21-35; DOI https://doi.org/10.1007/s11538-016-0212-y

LINKS

Fung Lam, Table of n, a(n) for n = 0..2000

E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, ECO method and hill-free generalized Motzkin paths, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.

FORMULA

G.f.: (1-z-q)/(z^2*(3-z-q)), where q = sqrt(1-2*z-3*z^2).

a(n) = sum(k=1..(n+3)/2, (k*sum(j=0..n-k+3, binomial(j,n-j+3)*binomial(n-k+3,j)))/(n-k+3)*(-1)^(k-1)). - Vladimir Kruchinin, Oct 22 2011

G.f.: 1/(1-z-z^3*M-z^4*M^2), where M is the g.f. of the Motzkin Numbers. - José Luis Ramírez Ramírez, Jan 28 2013

Recurrence: 2*(n+2)*a(n) = 3*(n-1)*a(n-4) + (4-n)*a(n-3) + 3*(n-3)*a(n-2) + (5*n+4)*a(n-1). - Fung Lam, Feb 03 2014

Asymptotics: a(n) ~ 3^(n+4)/(2^5*sqrt(3*Pi*n^3)). - Fung Lam, Mar 31 2014

EXAMPLE

a(4)=5 because the Motzkin paths of length 4 with no peaks at level 1 are:

HHHH, HUHD, UHDH, UHHD and UUDD, where H=(1,0), U=(1,1) and D=(1,-1).

MATHEMATICA

CoefficientList[Series[(1-x-Sqrt[1-2*x-3*x^2])/(x^2*(3-x-Sqrt[1-2*x-3*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 31 2014 *)

PROG

(Maxima)

a(n):=sum((k*sum(binomial(j, n-j+3)*binomial(n-k+3, j), j, 0, n-k+3))/(n-k+3)*(-1)^(k-1), k, 1, (n+3)/2); /* Vladimir Kruchinin, Oct 22 2011 */

CROSSREFS

Cf. A001006.

Sequence in context: A307788 A025273 A217333 * A036671 A152171 A132807

Adjacent sequences:  A089369 A089370 A089371 * A089373 A089374 A089375

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Dec 27 2003

STATUS

approved

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Last modified June 15 18:33 EDT 2021. Contains 345049 sequences. (Running on oeis4.)