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A109262 A Catalan transform of the Fibonacci numbers. 9
0, 1, 2, 6, 19, 63, 215, 749, 2650, 9490, 34318, 125104, 459152, 1694914, 6287896, 23429158, 87635243, 328917615, 1238303243, 4674847097, 17692789741, 67114622451, 255120892105, 971649360211, 3707176155659, 14167390221873 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A column of A109267.

Hankel transform is -F(2n). a(n+1) has Hankel transform F(2n+1). - Paul Barry, Nov 22 2007

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

FORMULA

G.f.: xc(x)/(1-xc(x)-x^2c(x)^2)=(1-sqrt(1-4x))/(2(sqrt(1-4x)+x)) where c(x) is the g.f. of A000108; a(n)=sum{k=0..n, (k/(2n-k))binomial(2n-k, n-k)F(k)}.

a(n)=Sum_{k, 0<=k<=n} A106566(n,k)*A000045(k). [From Philippe Deléham, Oct 28 2008]

a(n)=Sum_{k, 0<=k<=n} A039599(n,k)*(-1)^(k+1)*A000045(k). [From Philippe Deléham, Oct 28 2008]

Conjecture: n*a(n) +(-7*n+4)*a(n-1) +(7*n-2)*a(n-2) +(19*n-60)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 26 2012

Recurrence: n*(5*n-11)*a(n) = 2*(20*n^2 - 59*n + 30)*a(n-1) - 15*(5*n^2 - 19*n + 16)*a(n-2) - 2*(2*n-5)*(5*n-6)*a(n-3). - Vaclav Kotesovec, Feb 13 2014

a(n) ~ 5*4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

MATHEMATICA

CoefficientList[Series[(1-Sqrt[1-4*x])/(2*(Sqrt[1-4*x]+x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

CROSSREFS

Cf. A081696.

Sequence in context: A120900 A059712 A059713 * A006724 A057409 A141771

Adjacent sequences:  A109259 A109260 A109261 * A109263 A109264 A109265

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jun 24 2005

STATUS

approved

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Last modified November 28 19:08 EST 2014. Contains 250399 sequences.