A138364 The number of Motzkin n-paths with exactly one flat step.

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

Offset: 0

Author

Andrew V. Sutherland, Mar 16 2008

Keywords

• easy
• nonn
• changed

Comments

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

Examples

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• 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)

Crossrefs

Cf. A001700, A057977, A126869.

Programs

Magma

&cat[[0, Binomial(n, (n+1) div 2)]: n in [1..50 by 2]]; // Vincenzo Librandi, Apr 20 2015

Mathematica

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

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

Python

from math import comb
def A138364(n): return comb(n,n>>1) if n&1 else 0 # Chai Wah Wu, Aug 25 2025

Sage

def A138364(n):
    if is_even(n): return 0
    return binomial(n,n//2)
[A138364(n) for n in (0..42)]  # Peter Luschny, Mar 18 2014

Formula

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) + A126120(n) = A057977(n). - Peter Luschny, Mar 18 2014

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

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

Extensions

New name is a comment by David Scambler, May 02 2013. - Peter Luschny, Mar 18 2014