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A138364
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The number of Motzkin n-paths with exactly one flat step.
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39
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0, 1, 0, 3, 0, 10, 0, 35, 0, 126, 0, 462, 0, 1716, 0, 6435, 0, 24310, 0, 92378, 0, 352716, 0, 1352078, 0, 5200300, 0, 20058300, 0, 77558760, 0, 300540195, 0, 1166803110, 0, 4537567650, 0, 17672631900, 0, 68923264410, 0, 269128937220, 0
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OFFSET
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0,4
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COMMENTS
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An aerated version of A001700, which is the main entry for this sequence.
Number of paths in the half-plane x>=0, from (0,0) to (n,1), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=3, we have the 3 paths: UUD, UDU, DUU. - José Luis Ramírez Ramírez, Apr 19 2015
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REFERENCES
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Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1999.
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LINKS
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Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic Curves, L-Polynomials, and Random Matrices. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)
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FORMULA
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a(n) = binomial(n,(n+1)/2) for n odd, 0 otherwise.
E.g.f.: I_1(2z), where I_1 is the hyperbolic Bessel function of order 1.
a(n) = (1/(2*Pi))*integral(x=-2..2, x^n*x/sqrt((2+x)*(2-x))). - Peter Luschny, Sep 12 2011
G.f.: -(sqrt(1-4*x^2)+2*x^2-1)/(x*sqrt(1-4*x^2)+4*x^3-x). - Vladimir Kruchinin, Mar 08 2013
a(n) = Integral_[-Pi,Pi] cos^(n+1)/(2^(n-1)*Pi). - M. F. Hasler, Jul 12 2018
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EXAMPLE
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a(5)=10 since the coefficient of z^5 in I_1(2z) is binomial(5,3)=10.
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ n! BesselI[ 1, 2 x], {x, 0, n}]; (* Michael Somos, Mar 19 2014 *)
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PROG
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(PARI) x='x+O('x^66); concat([0], Vec( -(sqrt(1-4*x^2)+2*x^2-1) / (x*sqrt(1-4*x^2)+4*x^3-x))) \\ Joerg Arndt, May 08 2013
(Sage)
if is_even(n): return 0
return binomial(n, n//2)
(Magma) &cat[[0, Binomial(n, (n+1) div 2)]: n in [1..50 by 2]]; // Vincenzo Librandi, Apr 20 2015
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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