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%I #22 Sep 08 2022 08:45:17
%S 1,4,8,32,128,576,2688,13056,65024,330752,1710080,8962048,47497216,
%T 254132224,1370849280,7447117824,40707293184,223731253248,
%U 1235630948352,6853893292032,38166664839168,213288826699776,1195775593807872
%N Expansion of (1 - sqrt(1 - 4*x - 12*x^2))/(2*x).
%C 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).
%C Hankel transform is 4^C(n+1,2)*A128018(n). [_Paul Barry_, Nov 20 2009]
%C 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
%H Vincenzo Librandi, <a href="/A103970/b103970.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1 - sqrt(1-4*x*(1+3*x))/(2*x).
%F a(n) = Sum_{k=0..n} 3^(n-k)*C(k)*C(k+1, n-k).
%F 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
%p 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
%p taylor(((1-(1-4*z-12*z^2)^0.5)/(2*z)),z=0,32); # _Richard Choulet_, Dec 17 2009
%t CoefficientList[Series[(1 - Sqrt[1-4x-12x^2])/(2x), {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 18 2017 *)
%o (PARI) my(x='x+O('x^35)); Vec((1-sqrt(1-4*x-12*x^2))/(2*x)) \\ _G. C. Greubel_, Mar 16 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( (1-Sqrt(1-4*x-12*x^2))/(2*x) )); // _G. C. Greubel_, Mar 16 2019
%o (Sage) ((1-sqrt(1-4*x-12*x^2))/(2*x)).series(x, 35).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 16 2019
%Y Cf. A025227, A025229, A103971, A103972.
%Y Cf. A000108, A025228, A025230, A025231, A025232. [_Richard Choulet_, Dec 17 2009]
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 23 2005
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