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The number of Motzkin n-paths with exactly one flat step.
39

%I #56 Sep 08 2022 08:45:33

%S 0,1,0,3,0,10,0,35,0,126,0,462,0,1716,0,6435,0,24310,0,92378,0,352716,

%T 0,1352078,0,5200300,0,20058300,0,77558760,0,300540195,0,1166803110,0,

%U 4537567650,0,17672631900,0,68923264410,0,269128937220,0

%N The number of Motzkin n-paths with exactly one flat step.

%C An aerated version of A001700, which is the main entry for this sequence.

%C 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

%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Ch. 49, Hemisphere Publishing Corp., 1999.

%H G. C. Greubel, <a href="/A138364/b138364.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Kiran S. Kedlaya and Andrew V. Sutherland, <a href="http://dspace.mit.edu/handle/1721.1/64701">Hyperelliptic Curves, L-Polynomials, and Random Matrices</a>. In: Arithmetic, Geometry, Cryptography, and Coding Theory: International Conference, November 5-9, 2007, CIRM, Marseilles, France. (Contemporary Mathematics; v.487)

%F a(n) = binomial(n,(n+1)/2) for n odd, 0 otherwise.

%F E.g.f.: I_1(2z), where I_1 is the hyperbolic Bessel function of order 1.

%F a(n) = (1/(2*Pi))*integral(x=-2..2, x^n*x/sqrt((2+x)*(2-x))). - _Peter Luschny_, Sep 12 2011

%F 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

%F a(n) + A126120(n) = A057977(n). - _Peter Luschny_, Mar 18 2014

%F G.f.: z*C(z^2)/(1-2*z^2*C(z^2)), where C(z) is the g.f. of Catalan numbers. - _José Luis Ramírez Ramírez_, Apr 19 2015

%F a(n) = Integral_[-Pi,Pi] cos^(n+1)/(2^(n-1)*Pi). - _M. F. Hasler_, Jul 12 2018

%e a(5)=10 since the coefficient of z^5 in I_1(2z) is binomial(5,3)=10.

%t a[ n_] := SeriesCoefficient[ n! BesselI[ 1, 2 x], {x, 0, n}]; (* _Michael Somos_, Mar 19 2014 *)

%o (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

%o (Sage)

%o def A138364(n):

%o if is_even(n): return 0

%o return binomial(n,n//2)

%o [A138364(n) for n in (0..42)] # _Peter Luschny_, Mar 18 2014

%o (Magma) &cat[[0, Binomial(n, (n+1) div 2)]: n in [1..50 by 2]]; // _Vincenzo Librandi_, Apr 20 2015

%Y Cf. A001700, A057977, A126869.

%K easy,nonn

%O 0,4

%A _Andrew V. Sutherland_, Mar 16 2008

%E New name is a comment by _David Scambler_, May 02 2013. - _Peter Luschny_, Mar 18 2014