A101850 A Catalan transform of Pell(n+1).

1, 2, 7, 26, 100, 392, 1555, 6218, 25006, 100988, 409162, 1661948, 6764194, 27575732, 112570675, 460058906, 1881978694, 7704907724, 31566153058, 129400608044, 530734613920, 2177792579072, 8939838222718, 36711025334948

Offset: 0

Author

Paul Barry, Dec 18 2004

Keywords

• easy
• nonn

Comments

A Catalan transform of the Pell numbers A000129(n+1) under the mapping G(x) -> G(x*c(x)), c(x) the g.f. of A000108. The inverse mapping is H(x) -> H(x*(1-x)).

Hankel transform is 3^n. - Paul Barry, Jan 19 2009

Row sums of the Riordan matrix (1/(x+sqrt(1-4*x)),(1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x)))) (A188513). - Emanuele Munarini, Apr 02 2011

Equals the INVERT transform of A026641: (1, 1, 4, 13, 46, 166, ...). Example: a(4) = 100 = (1, 1, 2, 7, 26) dot (46, 13, 4, 1, 1) = (46 + 13 + 8 + 7 + 26 ) = 100. - Gary W. Adamson, Jan 10 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
• Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012

Crossrefs

Cf. A081696, A026641, A188513.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2/(3*Sqrt(1-4*x)+2*x-1) )); // G. C. Greubel, Feb 18 2019

Mathematica

CoefficientList[Series[(1-2x+3Sqrt[1-4x])/(4-16x-2x^2),{x,0,24}],x] (* Emanuele Munarini, Apr 02 2011 *)
CoefficientList[Series[2/(3*Sqrt[1-4*x]+2*x-1), {x,0,30}], x] (* or *) RecurrenceTable[{2*(n+3)*a[n+3] -4*(4*n+9)*a[n+2]+(31*n+45)*a[n+1] + 2*(2*n+3)*a[n]==0, a[0]==1, a[1]==2, a[2]==7}, a, {n, 0, 30}](* G. C. Greubel, Feb 18 2019 *)

Maxima

makelist(sum(binomial(2*n+1,n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2),k,0,n),n,0,12); /* Emanuele Munarini, Apr 02 2011 */

PARI

my(x='x+O('x^30)); Vec(2/(3*sqrt(1-4*x)+2*x-1)) \\ G. C. Greubel, Feb 18 2019

Sage

(2/(3*sqrt(1-4*x)+2*x-1)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

Formula

G.f.: 2/(3*sqrt(1-4*x) + 2*x - 1).

a(n) = Sum_{k=0..n} (k/(2*n-k))*binomial(2*n-k, n-k)*A000129(k+1).

a(n) = Sum_{k=0..n} A039599(n,k)*A016116(k). - Philippe Deléham, Oct 29 2008

G.f.: 1/(1-2*x-3*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-... (continued fraction). - Paul Barry, Jan 19 2009

From Emanuele Munarini, Apr 02 2011: (Start)

a(n) = [x^n] (1-2*x)/((1-2*x-x^2)(1-x)^(n+1)).

a(n) = Sum_{k=0..n} binomial(2*n+1, n+2*k+1)*2^(k+1)*(2*k+1)/(n+2*k+2).

Recurrence: 2*(n+3)*a(n+3) -4*(4*n+9)*a(n+2) +(31*n+45)*a(n+1) +2*(2*n+3)*a(n) = 0. (End)

From Gary W. Adamson, Jul 14 2011: (Start)

a(n) = the upper left term in M^n, M = an infinite square production matrix as follows:

2, 3, 0, 0, 0, 0, ...

1, 1, 1, 0, 0, 0, ...

1, 1, 1, 1, 0, 0, ...

1, 1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, 1, ...

... (End)

a(n) ~ (1/4)*(2 + 3/sqrt(2))^n. - Vaclav Kotesovec, Oct 17 2012