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).
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
Offset: 0
Keywords
Examples
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.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Maple
q := sqrt(1-4*z^2): g := 2*z^2*(1-q)/(q*(1-2*z+q)^2): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
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Mathematica
CoefficientList[Series[2*x^2*(1-Sqrt[1-4*x^2])/(Sqrt[1-4*x^2]*(1-2*x+ Sqrt[1-4*x^2])^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) -
PARI
x='x+O('x^50); concat([0,0,0,0], Vec(2*x^2*(1-sqrt(1-4*x^2))/(sqrt(1-4*x^2)*(1-2*x+ sqrt(1-4*x^2))^2))) \\ G. C. Greubel, Mar 26 2017
Formula
a(2*n+1) = 2*a(2*n).
a(2*n+4) = A029760(n).
G.f.: g = 2*z^2*(1-q)/(q*(1-2*z+q)^2), where q=sqrt(1-4*z^2).
a(n) ~ 2^(n-3/2)*sqrt(n)/sqrt(Pi) * (1-sqrt(2*Pi/n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: n*(n-4)*a(n) +(n^2-10*n+15)*a(n-1) +2*(-5*n^2+28*n-27)*a(n-2) -4*(n-3)*(n-8) *a(n-3) +24*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 14 2016
Comments