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

0, 0, 0, 1, 3, 8, 19, 43, 96, 206, 447, 936, 1998, 4128, 8718, 17865, 37446, 76322, 159079, 323020, 670350, 1357496, 2807370, 5673526, 11699768, 23607548, 48567174, 97877248, 200954796, 404584032, 829226364, 1668147573, 3413853906, 6863065482, 14026671159, 28182987108

Offset

0,5

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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

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).

a(n) ~ 2^(n-5/2)*sqrt(n)/sqrt(Pi) * (1 + 3*sqrt(Pi)/sqrt(2*n)). - Vaclav Kotesovec, Mar 21 2014

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

Example

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).

Maple

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);

Mathematica

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 *)

Prog

(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

Crossrefs

Sequence in context: A102712 A054480 A371796 * A347310 A332719 A326599

Adjacent sequences: A191784 A191785 A191786 * A191788 A191789 A191790

Keyword

nonn

Author

Emeric Deutsch, Jun 18 2011

Status

approved