A191319 Sum of pyramid weights of all dispersed Dyck paths of length n (i.e., of all Motzkin paths of length n with no (1,0) steps at positive heights).

0, 0, 1, 2, 7, 14, 37, 74, 177, 354, 807, 1614, 3579, 7158, 15591, 31182, 67071, 134142, 285861, 571722, 1209641, 2419282, 5089517, 10179034, 21314453, 42628906, 88918353, 177836706, 369734553, 739469106, 1533115953, 3066231906, 6341759073, 12683518146

Offset: 0

Author

Emeric Deutsch, Jun 01 2011

Keywords

• nonn

Comments

A pyramid in a dispersed Dyck path is a factor of the form U^h D^h, h being the height of the pyramid and U=(1,1), D=(1,-1). A pyramid in a dispersed Dyck path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a dispersed Dyck path is the sum of the heights of its maximal pyramids.

Examples

			a(4)=7 because the sum of the pyramid weights of HHHH, HH(UD), H(UD)H, (UD)HH, (UD)(UD), and (UUDD) is 0+1+1+1+2+2=7; the maximal pyramids are shown between parentheses.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.

Crossrefs

Cf. A191318.

Programs

Maple

g := z^2/((1-2*z)*(1-z^2)*sqrt(1-4*z^2)): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 33);

Mathematica

CoefficientList[Series[x^2/((1-2*x)*(1-x^2)*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PARI

Vec(1/((1-2*z)*(1-z^2)*sqrt(1-4*z^2))) \\ Charles R Greathouse IV, Jan 27 2016

Formula

a(n) = Sum_{k=0..n} k*A191318(n,k).

G.f.: g(z) = z^2/((1-2*z)*(1-z^2)*sqrt(1-4*z^2)).

a(n) ~ 2^(n+1/2)*sqrt(n)/(3*sqrt(Pi)) * (1 - 5/(6*n) + 1/4*(-1)^n/n). - Vaclav Kotesovec, Mar 20 2014

Conjecture: (-n+2)*a(n) +2*a(n-1) +5*(n-2)*a(n-2) -2*a(n-3) +4*(-n+2)*a(n-4)=0. - R. J. Mathar, Dec 07 2017