OFFSET
0,2
COMMENTS
Image of c(x), the g.f. of the Catalan numbers A000108 under the mapping g(x) -> (1+3x)g(x(1+3x)). In general, the image of the Catalan numbers under the mapping g(x) -> (1+i*x)g(x(1+i*x)) is given by a(n) = Sum_{k=0..n} i^(n-k)*C(k)*C(k+1,n-k).
Hankel transform is 4^C(n+1,2)*A128018(n). [Paul Barry, Nov 20 2009]
By following L. Comtet [Analyse Combinatoire Tomes 1 et 2, PUF, Paris 1970], we also obtain (n+1)*C(n) - 2*a*(2*n-1)*C(n-1) + 4*(n-2)*(a^2-b)*C(n-2) = 0. In the present case, we also have the asymptotic result: a(n) ~ sqrt(4/3)*2^(n-1)*3^(n+1)/sqrt(Pi*n^3) for large n. - Richard Choulet, Dec 17 2009
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (1 - sqrt(1-4*x*(1+3*x))/(2*x).
a(n) = Sum_{k=0..n} 3^(n-k)*C(k)*C(k+1, n-k).
D-finite with recurrence: (n+1)*a(n) = 2*(2*n-1)*a(n-1) + 12*(n-2)*a(n-2). - Richard Choulet, Dec 17 2009
MAPLE
n:=30:a(0):=1:a(1):=4: k:=1: for k from 1 to n do a(k+1):=sum('a(p)*a(k-p)', 'p'=0..k):od:seq(a(k), k=0..n); # Richard Choulet, Dec 17 2009
taylor(((1-(1-4*z-12*z^2)^0.5)/(2*z)), z=0, 32); # Richard Choulet, Dec 17 2009
MATHEMATICA
CoefficientList[Series[(1 - Sqrt[1-4x-12x^2])/(2x), {x, 0, 33}], x] (* Vincenzo Librandi, Aug 18 2017 *)
PROG
(PARI) my(x='x+O('x^35)); Vec((1-sqrt(1-4*x-12*x^2))/(2*x)) \\ G. C. Greubel, Mar 16 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( (1-Sqrt(1-4*x-12*x^2))/(2*x) )); // G. C. Greubel, Mar 16 2019
(Sage) ((1-sqrt(1-4*x-12*x^2))/(2*x)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Mar 16 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Feb 23 2005
STATUS
approved