A109262 A Catalan transform of the Fibonacci numbers.

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

Offset

0,3

Comments

A column of A109267.

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

Links

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

Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

Stoyan Dimitrov, On permutation patterns with constrained gap sizes, arXiv:2002.12322 [math.CO], 2020.

Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.

Guo-Niu Han, Enumeration of Standard Puzzles

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

Merve Taştan and Engin Özkan, Catalan transform of the k-Jacobsthal sequence, Electronic Journal of Mathematical Analysis and Applications (2020) Vol. 8, No. 2, 70-74.

Formula

G.f.: x*c(x)/(1 - x*c(x) - x^2*c(x)^2) = (1 - sqrt(1-4*x))/(2*(x + sqrt(1-4*x))) where c(x) is the g.f. of A000108.

a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*Fibonacci(k).

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

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

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

a(n) = (1/(2*sqrt(5)))*Catalan(n-1)*Sum_{j=0..1} ((-1)^j + sqrt(5)) * Hypergeometric2F1([2,1-n], [2*(1-n)], (1+(-1)^j*sqrt(5))/2). - G. C. Greubel, May 30 2022

Mathematica

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

Prog

(Magma) [n eq 0 select 0 else (&+[k*Binomial(2*n-k-1, n-1)*Fibonacci(k): k in [0..n]])/n: n in [0..30]]; // G. C. Greubel, May 30 2022
(SageMath) [0]+[(1/n)*sum(k*binomial(2*n-k-1, n-1)*fibonacci(k) for k in (1..n)) for n in (1..30)] # G. C. Greubel, May 30 2022

Crossrefs

Cf. A000045, A000108, A039599, A081696, A106566, A109267.

Sequence in context: A284216 A059712 A059713 * A006724 A057409 A346157

Adjacent sequences: A109259 A109260 A109261 * A109263 A109264 A109265

Keyword

easy,nonn

Author

Paul Barry, Jun 24 2005

Status

approved