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A191782 Sum of the lengths of the first ascents in all n-length left factors of Dyck paths. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n)=Sum(k*A191781(n,k), k>=0).

LINKS

Table of n, a(n) for n=1..35.

FORMULA

G.f.: g(z) = 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

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);

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 *)

CROSSREFS

A191781.

Sequence in context: A225198 A225199 A000219 * A027999 A005196 A032287

Adjacent sequences:  A191779 A191780 A191781 * A191783 A191784 A191785

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 18 2011

STATUS

approved

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Last modified December 8 22:51 EST 2016. Contains 278957 sequences.