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

0,3

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A344191 A032443 A180907 * A270561 A259858 A359711

Adjacent sequences: A143461 A143462 A143463 * A143465 A143466 A143467

Keyword

nonn

Author

Sergio Falcon, Oct 24 2008

Extensions

Offset corrected by Philippe Deléham, Oct 28 2008

Status

approved