OFFSET
0,2
COMMENTS
Note that while a(n) is even (for n > 0), it is a multiple of 4 except when n = 2^m-1, i.e., when Catalan(n) is odd.
Result of applying the Riordan matrix ((1+sqrt(1-4x))/2, (1-sqrt(1-4x))/2) (inverse of (1/(1-x), x(1-x)) to 3^n. - Paul Barry, Mar 12 2005
Hankel transform is A001787(n+1). - Paul Barry, Mar 15 2010
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
a(n) = A067337(2n, n).
G.f.: (1+sqrt(1-4x))/(3*sqrt(1-4x)-1). - Paul Barry, Mar 12 2005
G.f.: (1-x*c(x))/(1-3*x*c(x)), where c(x) is the g.f. of A000108. - Paul Barry, Mar 15 2010
Conjecture: 2*n*a(n) + (-17*n+12)*a(n-1) + 18*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 30 2012
G.f.: 1 + 2*x/(Q(0)-3*x), where Q(k) = 2*x + (k+1)/(2*k+1) - 2*x*(k+1)/(2*k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 03 2013
a(n) ~ 3^(2*n-1) / 2^n. - Vaclav Kotesovec, Feb 13 2014
EXAMPLE
a(2) = 2*9/2 - 1 = 8;
a(3) = 8*9/2 - 2 = 34;
a(4) = 34*9/2 - 5 = 148;
a(5) = 148*9/2 - 14 = 652.
MATHEMATICA
CoefficientList[Series[(1+Sqrt[1-4*x])/(3*Sqrt[1-4*x]-1), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, Jan 15 2002
STATUS
approved