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

1,2

Links

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

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A191781.

Sequence in context: A225199 A000219 A356941 * A358905 A027999 A005196

Adjacent sequences: A191779 A191780 A191781 * A191783 A191784 A191785

Keyword

nonn

Author

Emeric Deutsch, Jun 18 2011

Status

approved