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