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A191782
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Sum of the lengths of the first ascents in all n-length left factors of Dyck paths.
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2
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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
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OFFSET
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1,2
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..1000
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A191781.
Sequence in context: A225198 A225199 A000219 * A027999 A005196 A350851
Adjacent sequences: A191779 A191780 A191781 * A191783 A191784 A191785
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Jun 18 2011
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STATUS
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approved
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