OFFSET
0,3
COMMENTS
Hankel transform of a(n+1) is (1,1,1,...).
Hankel transform of a(n+2) is A001653(n+1) with g.f. (5-x)/(1-6x+x^2).
a(n+1) is the fourth binomial transform of the Catalan numbers A000108. - Paul Barry, Oct 09 2010
a(n) is the number of Schroeder paths of semilength n in which the H=(2,0) steps come in 4 colors and having no (2,0)-steps at levels 1,3,5,.... - José Luis Ramírez Ramírez, Mar 30 2013
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = 0^n/2 - A104497(n)/2.
From Paul Barry, Oct 09 2010: (Start)
a(n+1) = (1/Pi)*Integral_{x=4..8} x^n*sqrt(8-x)/(2*sqrt(x-4));
a(n+1) = 4^n*F(-n,1/2;2;-1). (End)
From Gary W. Adamson, Jul 21 2011: (Start)
a(n) = upper left term of M^(n-1), M = an infinite square production matrix as follows:
5, 1, 0, 0, ...
1, 5, 1, 0, ...
1, 1, 5, 1, ...
1, 1, 1, 5, ...
... (End)
Recurrence: n*a(n) = 2*(6*n-7)*a(n-1) - 32*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 2^(3*n-3/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
From Peter Bala, Feb 04 2024: (Start)
a(n+1) = Sum_{k = 0..n} binomial(n, k)* Catalan(k) * 4^(n-k).
a(n+1) = 4^n * hypergeom([-n, 1/2], [2], -1). (End)
MAPLE
seq(add(binomial(n-1, k)* (2*k)!/((k+1)*k!^2) * 4^(n-k-1), k = 0..n-1), n = 0..20); # Peter Bala, Feb 04 2024
MATHEMATICA
CoefficientList[Series[1/2*(1-Sqrt[1-8*x]/Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
PROG
(PARI) x='x+O('x^66); concat([0], Vec((1-sqrt(1-8*x)/sqrt(1-4*x))/2 )) /* Joerg Arndt, Mar 31 2013 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 11 2005, Mar 07 2008
STATUS
approved