A191782 Sum of the lengths of the first ascents in all n-length left factors of Dyck paths.

1, 3, 6, 13, 24, 49, 90, 181, 335, 671, 1253, 2507, 4718, 9437, 17874, 35749, 68067, 136135, 260337, 520675, 999361, 1998723, 3848221, 7696443, 14857999, 29715999, 57500459, 115000919, 222981434, 445962869, 866262914, 1732525829, 3370764539, 6741529079, 13135064249

Offset: 1

Author

Emeric Deutsch, Jun 18 2011

Keywords

• nonn

Examples

			a(4)=13 because in UDUD, UDUU, UUDD, UUDU, UUUD, and UUUU the sum of the lengths of the first ascents is 1 + 1 + 2 + 2 + 3 + 4  = 13.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A191781.

Programs

Maple

c := ((1-sqrt(1-4*z^2))*1/2)/z^2: g := z*c*(1+z*c^2)/((1-z)*(1-z*c)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 1 .. 35);
# Alternative:
a := n -> binomial(n+2, iquo(n,2)+1) - binomial(n, iquo(n,2)) - 1:
seq(a(n), n=1..35); # Peter Luschny, Feb 10 2019

Mathematica

Rest[With[{c=(1-Sqrt[1-4x^2])/(2x^2)},CoefficientList[ Series[ (x c (1+x c^2))/((1-x)(1-x c)),{x,0,40}],x]]] (* Harvey P. Dale, Jun 19 2011 *)

PARI

x='x+O('x^50); Vec(((1-sqrt(1-4*x^2))*(1-2*x^2+2*x^3-sqrt(1-4*x^2)))/(2*x^3*(1-x)*(2*x-1+sqrt(1-4*x^2)))) \\ G. C. Greubel, Mar 27 2017

Formula

a(n) = Sum_{k>=0} k*A191781(n,k).

G.f.: z*c*(1+z*c^2)/((1-z)*(1-z*c)), where c = (1-sqrt(1 -4*z^2)) / (2*z^2).

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

Conjecture: -(n+3)*(6*n-17)*a(n) +2*(6*n^2-2*n-57)*a(n-1) +3*(6*n^2-17*n+27)*a(n-2) -2*(2*n-3)*(12*n-25)*a(n-3) +4*(6*n-11)*(n-3)*a(n-4)=0. - R. J. Mathar, Jun 14 2016

a(n) = binomial(n+2, floor(n/2) + 1) - binomial(n, floor(n/2)) - 1. - Peter Luschny, Feb 10 2019