OFFSET
0,2
COMMENTS
The number of lattice paths consisting of 2*n steps either (1,1) or (1,-1) that return to the x-axis only at times that are a multiple of 4. - Peter Bala, Jan 02 2020
LINKS
Harry J. Smith, Table of n, a(n) for n=0..200
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
FORMULA
a(n) = 2^n*binomial(n, [n/2]);
G.f.: (sqrt((1+4*x)/(1-4*x))-1)/4/x. - Vladeta Jovovic, Apr 28 2003
E.g.f.: BesselI(0, 4*x)+BesselI(1, 4*x). - Vladeta Jovovic, Apr 28 2003
a(n) = 4^n*sum{k=0..n, C(n,k)C(k)/(-2)^k}, with C(n)=A000108(n). - Paul Barry, Dec 28 2006
(n+1)*a(n) -4*a(n-1) +16*(-n+1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
a(n) = (-4)^n*hypergeom([3/2,-n],[2],2). - Peter Luschny, Apr 26 2016
Sum_{n>=0} a(n)/6^n = 3/phi = A134973. - Peter McNair, Apr 30 2022
In general, for k>4, Sum_{n>=0} a(n)/k^n = (sqrt((k+4)/(k-4)) - 1) * k/4. - Vaclav Kotesovec, May 13 2022
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=0} 1/a(n) = 16*asin(1/4)/(3*sqrt(15)) + 4/3.
Sum_{n>=0} (-1)^n/a(n) = 4/5 - 16*asin(1/4)/(5*sqrt(15)). (End)
MATHEMATICA
Table[2^n Binomial[n, Floor[n/2]], {n, 0, 30}] (* Harvey P. Dale, Oct 15 2017 *)
PROG
(PARI) { for (n=0, 200, write("b060899.txt", n, " ", 2^n*binomial(n, n\2)); ) } \\ Harry J. Smith, Jul 14 2009
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David W. Wilson, May 05 2001
STATUS
approved