

A191309


Number of peaks at height >= 2 in all dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights).


2



0, 0, 0, 0, 1, 2, 8, 16, 47, 94, 244, 488, 1186, 2372, 5536, 11072, 25147, 50294, 112028, 224056, 491870, 983740, 2135440, 4270880, 9188406, 18376812, 39249768, 78499536, 166656772, 333313544, 704069248, 1408138496, 2961699667, 5923399334, 12412521388, 24825042776
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OFFSET

0,6


COMMENTS

Also number of valleys (i.e., DU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+1+0 = 1 valley.
Also number of doublerises (i.e., UU's) in all dispersed Dyck paths of length n. Example: a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 = 1 doublerise.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000


FORMULA

a(2*n+1) = 2*a(2*n).
a(2*n+4) = A029760(n).
G.f.: g = 2*z^2*(1q)/(q*(12*z+q)^2), where q=sqrt(14*z^2).
a(n) ~ 2^(n3/2)*sqrt(n)/sqrt(Pi) * (1sqrt(2*Pi/n)).  Vaclav Kotesovec, Mar 20 2014
Conjecture: n*(n4)*a(n) +(n^210*n+15)*a(n1) +2*(5*n^2+28*n27)*a(n2) 4*(n3)*(n8) *a(n3) +24*(n3)*(n4)*a(n4)=0.  R. J. Mathar, Jun 14 2016


EXAMPLE

a(4)=1 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD we have 0+0+0+0+0+1 =1 peak at height >=2.


MAPLE

q := sqrt(14*z^2): g := 2*z^2*(1q)/(q*(12*z+q)^2): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);


MATHEMATICA

CoefficientList[Series[2*x^2*(1Sqrt[14*x^2])/(Sqrt[14*x^2]*(12*x+ Sqrt[14*x^2])^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)


PROG

(PARI) x='x+O('x^50); concat([0, 0, 0, 0], Vec(2*x^2*(1sqrt(14*x^2))/(sqrt(14*x^2)*(12*x+ sqrt(14*x^2))^2))) \\ G. C. Greubel, Mar 26 2017


CROSSREFS

Cf. A029760, A191308.
Sequence in context: A176143 A296946 A096227 * A323351 A134353 A280229
Adjacent sequences: A191306 A191307 A191308 * A191310 A191311 A191312


KEYWORD

nonn


AUTHOR

Emeric Deutsch, May 30 2011


STATUS

approved



