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
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
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