A143464 Catalan transform of the Pell sequence.

0, 1, 3, 11, 42, 164, 649, 2591, 10408, 41998, 170050, 690370, 2808714, 11446642, 46715469, 190876527, 780679200, 3195628806, 13090353594, 53655587034, 220045073988, 902842397664, 3705876933930, 15216954519222, 62503485455208

Offset: 0

Author

Sergio Falcon, Oct 24 2008

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1000
• Paul Barry, A Catalan Transform and Related Transformations of Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.4
• Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
• Sergio Falcón, 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.
• Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals 2007; 33(1): 38-49.
• Sergio Falcón and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
• Merve Taştan and Engin Özkan, Catalan transform of the k-Pell, k-Pell-Lucas and modified k-Pell sequence, Notes on Num. Theory and Disc. Math. (2021) Vol. 27, No. 1, 198-207.

Crossrefs

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

Programs

Mathematica

a[n_]:= a[n]= If[n==0, 0, Sum[i*Binomial[2n-i,n-i]*Fibonacci[i,2]/(2n-i), {i,n}]];
Table[a[n], {n,0,30}] (* modified by G. C. Greubel, May 31 2022 *)

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

SageMath

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

Formula

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

From Philippe Deléham, Oct 28 2008: (Start)

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

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

G.f.: ((1+x)*sqrt(1-4*x) - (1-5*x))/(2*(2 - 8*x - x^2)). - Mark van Hoeij, May 01 2013

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

a(n) ~ (1 + sqrt(2))^(2*n - 1) / 2^(2 + n/2). - Vaclav Kotesovec, May 31 2022

Extensions

Offset corrected by Philippe Deléham, Oct 28 2008