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A103971
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Expansion of (1-sqrt(1-4x-16x^2))/(2x).
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1
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1, 5, 10, 45, 190, 930, 4660, 24445, 131190, 719830, 4013260, 22684370, 129661740, 748252580, 4353379560, 25508284445, 150392391590, 891549228430, 5310994644060, 31775749689670, 190860711108740, 1150473009844380
(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+4x)g(x(1+4x)). 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)}.
More generally, the sequence C for which C(0)=a, C(1)=b and C(n+1)=sum(C(k)*C(n-k),k=0..n) has the following G.f f: f(z)= (1-sqrt(1-4*z*(a-(a^2-b)*z)))/(2*z). We obtain: C(n)=(sum(-1)^(p-1)*2^{n-p}a^{n-2*p-1}*(a^2-b)^p*((2*n-2*p-1)*...*5*3*1/(p!*(n-2*p+1)!)),p=0..floor((n+1)/2)). By following 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. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 17 2009]
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FORMULA
| G.f. : (1-sqrt(1-4x(1+4x))/(2x); a(n)=sum{k=0..n, 4^(n-k)C(k)C(k+1, n-k)}.
Another recurrence formula: (n+1)*a(n)=2*(2*n-1)*a(n-1)+16*(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):=5: 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); [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 17 2009]
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CROSSREFS
| Cf. A000108, A025227, A025228, A025229, A025230, A025231, A025232. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Dec 17 2009]
Sequence in context: A187877 A122173 A083515 * A035406 A103932 A034190
Adjacent sequences: A103968 A103969 A103970 * A103972 A103973 A103974
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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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