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Catalan transform of the Pell sequence.
5

%I #36 Jul 16 2022 11:19:54

%S 0,1,3,11,42,164,649,2591,10408,41998,170050,690370,2808714,11446642,

%T 46715469,190876527,780679200,3195628806,13090353594,53655587034,

%U 220045073988,902842397664,3705876933930,15216954519222,62503485455208

%N Catalan transform of the Pell sequence.

%H Michael De Vlieger, <a href="/A143464/b143464.txt">Table of n, a(n) for n = 0..1000</a>

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations of Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4

%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012

%H Sergio Falcón, <a href="http://www.mathnet.or.kr/mathnet/thesis_file/CKMS-28-4-827-832.pdf">Catalan transform of the K-Fibonacci sequence</a>, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.

%H Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.10.022">The k-Fibonacci sequence and the Pascal 2-triangle</a>, Chaos, Solitons & Fractals 2007; 33(1): 38-49.

%H Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.09.022">On the Fibonacci k-numbers</a>, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.

%H Merve Taştan and Engin Özkan, <a href="https://doi.org/10.7546/nntdm.2021.27.1.198-207">Catalan transform of the k-Pell, k-Pell-Lucas and modified k-Pell sequence</a>, Notes on Num. Theory and Disc. Math. (2021) Vol. 27, No. 1, 198-207.

%F a(n) = Sum_{j=0..n} (j/(2*n-j))*binomial(2*n-j, n)*Pell(j), with a(0) = 0.

%F From _Philippe Deléham_, Oct 28 2008: (Start)

%F a(n) = Sum_{k=0..n} A106566(n,k)*A000129(k).

%F a(n) = Sum_{k=0..n} A039599(n,k)*A000035(k)*A016116(k). (End)

%F G.f.: ((1+x)*sqrt(1-4*x) - (1-5*x))/(2*(2 - 8*x - x^2)). - _Mark van Hoeij_, May 01 2013

%F a(n) = (1/(2*sqrt(2)))*Catalan(n-1)*Sum_{j=0..1} ((-1)^j + sqrt(2)) * Hypergeometric2F1([2,1-n], [2*(1-n)], 1+(-1)^j*sqrt(2)) - [n=0]/2. - _G. C. Greubel_, May 31 2022

%F a(n) ~ (1 + sqrt(2))^(2*n - 1) / 2^(2 + n/2). - _Vaclav Kotesovec_, May 31 2022

%t a[n_]:= a[n]= If[n==0, 0, Sum[i*Binomial[2n-i,n-i]*Fibonacci[i,2]/(2n-i), {i,n}]];

%t Table[a[n], {n,0,30}] (* modified by _G. C. Greubel_, May 31 2022 *)

%o (PARI) my(x='x+O('x^66)); concat([0],Vec((1-5*x-(1+x)*sqrt(1-4*x))/(2*x^2+16*x-4))) \\ _Joerg Arndt_, May 01 2013

%o (SageMath)

%o def Pell(n): return round( ((1+sqrt(2))^n - (1-sqrt(2))^n)/(2*sqrt(2)) )

%o [0]+[(1/n)*sum(k*binomial(2*n-k-1, n-1)*Pell(k) for k in (1..n)) for n in (1..30)] # _G. C. Greubel_, May 31 2022

%Y Cf. A000035, A000129, A016116, A039599, A106566, A109262.

%K nonn

%O 0,3

%A _Sergio Falcon_, Oct 24 2008

%E Offset corrected by _Philippe Deléham_, Oct 28 2008