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A191307 Sum of the heights of the first peaks in all dispersed Dyck paths of length n (i.e., in Motzkin paths of length n with no (1,0)-steps at positive heights). 2

%I #17 Mar 27 2017 15:43:03

%S 0,0,1,2,6,11,26,47,103,187,397,727,1519,2806,5809,10814,22254,41702,

%T 85460,161042,329002,622932,1269578,2413644,4909788,9367188,19024888,

%U 36408748,73850908,141714823,287137498,552320023,1118042743,2155201063,4359162493,8419091443

%N Sum of the heights of the first peaks in all dispersed Dyck paths of length n (i.e., in Motzkin paths of length n with no (1,0)-steps at positive heights).

%H G. C. Greubel, <a href="/A191307/b191307.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k>=0} k*A191306(n,k).

%F G.f.: ((1-z-z^2)*sqrt(1-4*z^2) - (1-2*z)*(1+z-z^2))/(2*z^3*(1-z)*(1-2*z)).

%F a(n) ~ 2^(n+3/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 20 2014

%F Conjecture: -(n+3)*(n-2)*a(n) +(n^2+3*n-6)*a(n-1) +2*n*(2*n-3)*a(n-2) - 4*n*(n-1)*a(n-3)=0. - _R. J. Mathar_, Jun 14 2016

%e a(4)=6 because, denoting U=(1,1), D=(1,-1), H=(1,0), in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD the sum of the heights of the first peaks is 0+1+1+1+1+2=6.

%p g:=(((1-z-z^2)*sqrt(1-4*z^2)-(1-2*z)*(1+z-z^2))*1/2)/(z^3*(1-z)*(1-2*z)): gser:=series(g,z=0,40): seq(coeff(gser,z,n),n=0..35);

%t CoefficientList[Series[(((1-x-x^2)*Sqrt[1-4*x^2]-(1-2*x)*(1+x-x^2))*1/2) /(x^3*(1-x)*(1-2*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)

%o (PARI) x='x+O('x^50); concat([0,0], Vec(((1-z-z^2)*sqrt(1-4*z^2) - (1-2*z)*(1+z-z^2))/(2*z^3*(1-z)*(1-2*z)))) \\ _G. C. Greubel_, Mar 26 2017

%Y Cf. A191306.

%K nonn

%O 0,4

%A _Emeric Deutsch_, May 30 2011

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)