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%I #43 Jun 01 2022 09:44:12
%S 0,1,2,6,19,63,215,749,2650,9490,34318,125104,459152,1694914,6287896,
%T 23429158,87635243,328917615,1238303243,4674847097,17692789741,
%U 67114622451,255120892105,971649360211,3707176155659,14167390221873
%N A Catalan transform of the Fibonacci numbers.
%C A column of A109267.
%C Hankel transform is -Fibonacci(2*n). a(n+1) has Hankel transform Fibonacci(2*n+1). - _Paul Barry_, Nov 22 2007
%H Vincenzo Librandi, <a href="/A109262/b109262.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%H Stoyan Dimitrov, <a href="https://arxiv.org/abs/2002.12322">On permutation patterns with constrained gap sizes</a>, arXiv:2002.12322 [math.CO], 2020.
%H Sergio Falcon, <a href="https://doi.org/10.4134/CKMS.2013.28.4.827">Catalan transform of the K-Fibonacci sequence</a>, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832.
%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H Merve Taştan and Engin Özkan, <a href="https://www.researchgate.net/publication/337102299_CATALAN_TRANSFORM_OF_THE_k-JACOBSTHAL_SEQUENCE">Catalan transform of the k-Jacobsthal sequence</a>, Electronic Journal of Mathematical Analysis and Applications (2020) Vol. 8, No. 2, 70-74.
%F 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.
%F a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*Fibonacci(k).
%F a(n) = Sum_{k=0..n} A106566(n,k)*A000045(k). - _Philippe Deléham_, Oct 28 2008
%F a(n) = Sum_{k=0..n} A039599(n,k)*(-1)^(k+1)*A000045(k). - _Philippe Deléham_, Oct 28 2008
%F 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
%F 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
%F a(n) ~ 5*4^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014
%F 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
%t CoefficientList[Series[(1-Sqrt[1-4*x])/(2*(Sqrt[1-4*x]+x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%o (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
%o (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
%Y Cf. A000045, A000108, A039599, A081696, A106566, A109267.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jun 24 2005
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