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A127360 a(n)=sum{k=0..n, C(n,floor(k/2))*4^(n-k)}. 4
1, 5, 22, 95, 406, 1730, 7360, 31295, 133030, 565430, 2403172, 10213670, 43408444, 184486580, 784069252, 3332296895, 14162266630, 60189642830, 255806000260 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Hankel transform is (-3)^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)/(1-4x) 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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

FORMULA

G.f.: (1/sqrt(1-4x^2))(1+x*c(x^2))/(1-4*x*c(x^2))

a(n)= Sum_{k, 0<=k<=n} A061554(n,k)*4^k. [From Philippe Deléham, Dec 04 2009]

Recurrence: 4*n*a(n) = (17*n + 8)*a(n-1) + 2*(8*n - 33)*a(n-2) - 68*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 5*17^n/4^(n+1). - Vaclav Kotesovec, Oct 19 2012

MATHEMATICA

CoefficientList[Series[(1/Sqrt[1-4x^2])*(1+x*(1-Sqrt[1-4*x^2])/(2*x^2))/(1-4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

CROSSREFS

Cf. A107430 [From Philippe Deléham, Sep 16 2009]

Sequence in context: A049675 A053154 A141222 * A116415 A026861 A026888

Adjacent sequences:  A127357 A127358 A127359 * A127361 A127362 A127363

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jan 11 2007

STATUS

approved

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Last modified February 21 23:44 EST 2019. Contains 320381 sequences. (Running on oeis4.)