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%I #24 Dec 07 2017 04:55:16
%S 0,0,1,2,7,14,37,74,177,354,807,1614,3579,7158,15591,31182,67071,
%T 134142,285861,571722,1209641,2419282,5089517,10179034,21314453,
%U 42628906,88918353,177836706,369734553,739469106,1533115953,3066231906,6341759073,12683518146
%N 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).
%C 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.
%H G. C. Greubel, <a href="/A191319/b191319.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.
%F a(n) = Sum_{k=0..n} k*A191318(n,k).
%F G.f.: g(z) = z^2/((1-2*z)*(1-z^2)*sqrt(1-4*z^2)).
%F 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
%F 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
%e 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.
%p 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);
%t CoefficientList[Series[x^2/((1-2*x)*(1-x^2)*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) Vec(1/((1-2*z)*(1-z^2)*sqrt(1-4*z^2))) \\ _Charles R Greathouse IV_, Jan 27 2016
%Y Cf. A191318.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Jun 01 2011
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