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A166302 Sum of pyramid weights of all Dyck paths of semilength n that have no ascents and no descents of length 1. 2
0, 0, 2, 3, 8, 19, 44, 106, 257, 628, 1549, 3844, 9588, 24020, 60391, 152298, 385085, 975904, 2478129, 6303861, 16060946, 40977605, 104682165, 267730426, 685451776, 1756593392, 4505537267, 11565724164, 29711413595, 76379060176, 196473781247 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word 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 Dyck path (word) 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*A166301(n,k).

G.f.: z*(2-z)*[1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2))]/[2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))].

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

EXAMPLE

a(5)=19 because the pyramid weights of (UUDD)(UUUDDD), (UUUDDD)(UUDD), U(UUDD)(UUDD)D, and (UUUUUDDDDD) are 5, 5, 4, and 5, respectively (the maximal pyramids are shown between parentheses).

MAPLE

G := (1/2)*z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/((1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);

MATHEMATICA

CoefficientList[Series[1/2*x*(2-x)*(1+x-x^2-Sqrt[(1+x+x^2)*(1-3*x+x^2)]) /((1-x)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

PROG

(PARI) z='z+O('z^50); concat([0, 0], Vec(z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/(2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))))) \\ G. C. Greubel, Mar 22 2017

CROSSREFS

Cf. A166301, A091866.

Sequence in context: A242099 A041205 A002356 * A100342 A041281 A078343

Adjacent sequences:  A166299 A166300 A166301 * A166303 A166304 A166305

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 07 2009

STATUS

approved

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Last modified October 16 14:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)