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A103970
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Expansion of (1-sqrt(1-4x-12x^2))/(2x).
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1
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1, 4, 8, 32, 128, 576, 2688, 13056, 65024, 330752, 1710080, 8962048, 47497216, 254132224, 1370849280, 7447117824, 40707293184, 223731253248, 1235630948352, 6853893292032, 38166664839168, 213288826699776, 1195775593807872
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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). [From Paul Barry (pbarry(AT)wit.ie), Nov 20 2009]
By following L. Comtet [Analyse Combinatoire Tomes 1 et 2, PUF, Paris 1970], we obtain also: (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 have also the asymptotic result: a(n)#(4/3)^0.5*2^(n-1)*3^(n+1)/(Pi*n^3)^0.5 for large n. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 17 2009]
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FORMULA
| G.f. : (1-sqrt(1-4x(1+3x))/(2x); a(n)=sum{k=0..n, 3^(n-k)C(k)C(k+1, n-k)}.
(n+1)*a(n)=2*(2*n-1)*a(n-1)+12*(n-2)*a(n-2). [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 17 2009]
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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); taylor(((1-(1-4*z-12*z^2)^0.5)/(2*z)), z=0, 32); [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 17 2009]
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CROSSREFS
| Cf. A025227, A025229, A103971, A103972.
Cf. A000108, A025228, A025230, A025231, A025232. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 17 2009]
Sequence in context: A086344 A068205 A113479 * A034785 A075398 A072868
Adjacent sequences: A103967 A103968 A103969 * A103971 A103972 A103973
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 23 2005
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