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

0,4

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.

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.

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A191318.

Sequence in context: A320651 A167762 A191389 * A018497 A178748 A194590

Adjacent sequences: A191316 A191317 A191318 * A191320 A191321 A191322

Keyword

nonn

Author

Emeric Deutsch, Jun 01 2011

Status

approved