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

0,4

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

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)

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A001700, A057977, A126869.

Sequence in context: A094472 A347999 A028850 * A095364 A094052 A161678

Adjacent sequences: A138361 A138362 A138363 * A138365 A138366 A138367

Keyword

easy,nonn

Author

Andrew V. Sutherland, Mar 16 2008

Extensions

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

Status

approved