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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). 3
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 01 2011
STATUS
approved

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Last modified February 24 06:47 EST 2024. Contains 370294 sequences. (Running on oeis4.)