

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
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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



FORMULA

a(n) = Sum_{k=0..n} k*A191318(n,k).
G.f.: g(z) = z^2/((12*z)*(1z^2)*sqrt(14*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(n1) +5*(n2)*a(n2) 2*a(n3) +4*(n+2)*a(n4)=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/((12*z)*(1z^2)*sqrt(14*z^2)): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 33);


MATHEMATICA

CoefficientList[Series[x^2/((12*x)*(1x^2)*Sqrt[14*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)


PROG



CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



