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A127359
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a(n)=sum{k=0..n, C(n,floor(k/2))*3^(n-k)}.
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4
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1, 4, 14, 48, 162, 544, 1820, 6080, 20290, 67680, 225684, 752448, 2508468, 8362176, 27875064, 92919168, 309734850, 1032458080, 3441543140, 11471842880
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OFFSET
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0,2
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COMMENTS
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Hankel transform is (-2)^n. In general, given r>=0, the sequence given by sum{k=0..n, C(n,floor(k/2))*r^(n-k)} has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-3x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
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LINKS
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Table of n, a(n) for n=0..19.
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FORMULA
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G.f.: (1/sqrt(1-4x^2))(1+x*c(x^2))/(1-3*x*c(x^2))
a(n)= Sum_{k, 0<=k<=n} A061554(n,k)*3^k. [From Philippe DELEHAM, Dec 04 2009]
Recurrence: 3*n*a(n) = 2*(5*n + 3)*a(n-1) + 4*(3*n - 11)*a(n-2) - 40*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ 4*10^n/3^(n+1). - Vaclav Kotesovec, Oct 19 2012
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MATHEMATICA
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Table[Sum[Binomial[n, Floor[k/2]]*3^(n-k), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 19 2012 *)
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CROSSREFS
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Cf. A107430 [From Philippe DELEHAM, Sep 16 2009]
Sequence in context: A022632 A027906 A047135 * A007070 A204089 A092489
Adjacent sequences: A127356 A127357 A127358 * A127360 A127361 A127362
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Jan 11 2007
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STATUS
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approved
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