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A097779 Number of Motzkin paths of length n, starting with an up step, ending with a down step and having no peaks (can be easily expressed using RNA secondary structure terminology). 2
1, 0, 0, 1, 1, 2, 5, 11, 25, 58, 135, 317, 750, 1785, 4272, 10275, 24823, 60210, 146576, 358010, 877087, 2154751, 5307166, 13102511, 32418806, 80375267, 199650310, 496803811, 1238276667, 3091173482, 7727893389, 19346109435, 48493869237 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Table of n, a(n) for n=0..32.

FORMULA

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

EXAMPLE

a(6)=5 because we have UHHHHD, UHDUHD, UUHHDD, UHUHDD and UUHDHD, where U=(1,1), D=(1,-1) and H=(1,0).

MAPLE

G:=z+1/2*(1-z)^2/z^2*(1-z+z^2-sqrt(1-2*z-z^2-2*z^3+z^4)): Gser:=series(G, z=0, 40): 1, seq(coeff(Gser, z^n), n=1..37);

MATHEMATICA

CoefficientList[Series[x+(1-x)^2 (1-x+x^2-Sqrt[1-2x-x^2-2x^3+x^4])/(2x^2), {x, 0, 40}], x] (* Harvey P. Dale, Dec 24 2016 *)

CROSSREFS

Cf. A004148.

Sequence in context: A094981 A304969 A239812 * A319768 A106336 A226974

Adjacent sequences:  A097776 A097777 A097778 * A097780 A097781 A097782

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Sep 11 2004

STATUS

approved

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Last modified November 12 19:25 EST 2019. Contains 329078 sequences. (Running on oeis4.)