cp's OEIS Frontend

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).

%I A191307 #17 Mar 27 2017 15:43:03
%S A191307 0,0,1,2,6,11,26,47,103,187,397,727,1519,2806,5809,10814,22254,41702,
%T A191307 85460,161042,329002,622932,1269578,2413644,4909788,9367188,19024888,
%U A191307 36408748,73850908,141714823,287137498,552320023,1118042743,2155201063,4359162493,8419091443
%N 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).
%H A191307 G. C. Greubel, <a href="/A191307/b191307.txt">Table of n, a(n) for n = 0..1000</a>
%F A191307 a(n) = Sum_{k>=0} k*A191306(n,k).
%F A191307 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 A191307 a(n) ~ 2^(n+3/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 20 2014
%F A191307 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 A191307 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 A191307 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 A191307 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 A191307 (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 A191307 Cf. A191306.
%K A191307 nonn
%O A191307 0,4
%A A191307 _Emeric Deutsch_, May 30 2011