A191396 Sum of the heights of the base pyramids in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0)-steps at positive heights). A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis. Here U=(1,1) and D=(1,-1).
0, 0, 1, 2, 7, 14, 35, 70, 156, 312, 663, 1326, 2756, 5512, 11325, 22650, 46227, 92454, 187891, 375782, 761465, 1522930, 3079475, 6158950, 12434015, 24868030, 50142687, 100285374, 202010787, 404021574, 813191039, 1626382078, 3271314744, 6542629488
Offset: 0
Keywords
Examples
a(4)=7 because in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD the sum of the heights of the base pyramids are 0, 1, 1, 1, 2, and 2, respectively.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Maple
g := 4*z^2/(1-2*z-z^2+2*z^3+(1-z^2)*sqrt(1-4*z^2))^2: gser := series(g, z = 0, 37): seq(coeff(gser, z, n), n = 0 .. 33);
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Mathematica
CoefficientList[Series[4*x^2/(1-2*x-x^2+2*x^3+(1-x^2)*Sqrt[1-4*x^2])^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *) -
PARI
x='x+O('x^50); concat([0,0], Vec(4*x^2/(1-2*x-x^2+2*x^3+(1-x^2)*sqrt(1-4*x^2))^2)) \\ G. C. Greubel, May 27 2017
Formula
a(n) = Sum_{k>=0} k*A191395(n,k).
G.f.: g(z) = 4*z^2/(1-2*z-z^2+2*z^3+(1-z^2)*sqrt(1-4*z^2))^2.
a(n) ~ 2^(n+3)/9 * (1-sqrt(2)/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014
D-finite with recurrence n*a(n) -2*n*a(n-1) +6*(-n+2)*a(n-2) +12*(n-2)*a(n-3) +3*(3*n-8)*a(n-4) +6*(-3*n+8)*a(n-5) +4*(-n+3)*a(n-6) +8*(n-3)*a(n-7)=0. - R. J. Mathar, Oct 08 2016