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A109188
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Number of (1,0) steps in all Grand Motzkin paths of length n.
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8
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1, 2, 9, 28, 95, 306, 987, 3144, 9963, 31390, 98483, 307836, 959257, 2981174, 9243405, 28601712, 88342659, 272428758, 838903371, 2579937060, 7924966749, 24317716038, 74546117121, 228317474952, 698708409525, 2136597743826
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OFFSET
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1,2
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COMMENTS
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A Grand Motzkin path is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).
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LINKS
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FORMULA
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G.f.: x*(1 - x)/(1 - 2*x - 3*x^2)^(3/2).
E.g.f.: a(n) = n! * [x^n] exp(x)*((1 + x)*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x)). - Peter Luschny, Aug 25 2012
D-finite with recurrence (-n+1)*a(n) + (3*n-4)*a(n-1) + (n+5)*a(n-2) + 3*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012
a(n) = n*hypergeom([1-n/2, 1/2-n/2], [1], 4) . - Peter Luschny, Sep 18 2014
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EXAMPLE
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a(3)=9 because we have the following 7 (=A002426(3)) Grand Motzkin paths of length 3: hhh, hud, hdu, udh, duh, uhd and dhu; they have a total of 9 h-steps.
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MAPLE
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g:=z*(1-z)/(1-2*z-3*z^2)^(3/2): gser:=series(g, z=0, 33): seq(coeff(gser, z^n), n=1..30);
a := n -> n*hypergeom([1-n/2, 1/2-n/2], [1], 4):
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MATHEMATICA
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Rest[CoefficientList[Series[x*(1-x)/(1-2*x-3*x^2)^(3/2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Sep 18 2014 *)
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PROG
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(PARI) Vec(z*(1-z)/(1-2*z-3*z^2)^(3/2) + O(z^50)) \\ G. C. Greubel, Jan 31 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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