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A191787 Number of triple-rises in all length n left factors of Dyck paths (triple-rise = three consecutive (1,1)-steps). 2

%I #14 Jul 24 2022 13:07:14

%S 0,0,0,1,3,8,19,43,96,206,447,936,1998,4128,8718,17865,37446,76322,

%T 159079,323020,670350,1357496,2807370,5673526,11699768,23607548,

%U 48567174,97877248,200954796,404584032,829226364,1668147573,3413853906,6863065482,14026671159,28182987108

%N Number of triple-rises in all length n left factors of Dyck paths (triple-rise = three consecutive (1,1)-steps).

%H G. C. Greubel, <a href="/A191787/b191787.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k>=0} k*A191785(n,k).

%F G.f.: (1-6*z^2-z^3+8*z^4+4*z^5-(1-4*z^2-z^3)*sqrt(1-4*z^2))/(2*z*(1+2*z)*(1-2*z)^2).

%F a(n) ~ 2^(n-5/2)*sqrt(n)/sqrt(Pi) * (1 + 3*sqrt(Pi)/sqrt(2*n)). - _Vaclav Kotesovec_, Mar 21 2014

%F D-finite with recurrence +(n+1)*(n^3-3*n^2-62*n+192)*a(n) -2*(n^4-2*n^3-81*n^2+186*n+192)*a(n-1) -4*(n^4-3*n^3-49*n^2+267*n-384)*a(n-2) +8*(n-3)*(n^3-65*n+128)*a(n-3)=0. - _R. J. Mathar_, Jun 14 2016

%e a(4)=3 because in UDUD, UDUU, UUDD, UUDU, (UUU)D, and (U[UU)U] we have a total of 0 + 0 + 0 + 0 +1 + 2 = 3 triple-rises (shown between parentheses).

%p G := ((1-6*z^2-z^3+8*z^4+4*z^5-(1-4*z^2-z^3)*sqrt(1-4*z^2))*1/2)/(z*(1+2*z)*(1-2*z)^2): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 35);

%t CoefficientList[Series[((1-6*x^2-x^3+8*x^4+4*x^5-(1-4*x^2-x^3)*Sqrt[1-4*x^2])*1/2)/(x*(1+2*x)*(1-2*x)^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)

%o (PARI) z='z+O('z^50); concat([0,0,0], Vec((1-6*z^2-z^3+8*z^4+4*z^5-(1-4*z^2-z^3)*sqrt(1-4*z^2))/(2*z*(1+2*z)*(1-2*z)^2))) \\ _G. C. Greubel_, Mar 27 2017

%K nonn

%O 0,5

%A _Emeric Deutsch_, Jun 18 2011

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