A104498 Expansion of (1/2)*(1-sqrt(1-8*x)/sqrt(1-4*x)).

0, 1, 5, 26, 141, 798, 4706, 28820, 182461, 1188406, 7926102, 53910828, 372671250, 2610977388, 18498911268, 132310178472, 953981219997, 6926326243110, 50593306470542, 371528742549692, 2741187564459910, 20310150708154564

Offset

0,3

Comments

Hankel transform of a(n+1) is (1,1,1,...).

Hankel transform of a(n+2) is A001653(n+1) with g.f. (5-x)/(1-6x+x^2).

a(n+1) is the fourth binomial transform of the Catalan numbers A000108. - Paul Barry, Oct 09 2010

a(n) is the number of Schroeder paths of semilength n in which the H=(2,0) steps come in 4 colors and having no (2,0)-steps at levels 1,3,5,.... - José Luis Ramírez Ramírez, Mar 30 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

a(n) = 0^n/2 - A104497(n)/2.

From Paul Barry, Oct 09 2010: (Start)

a(n+1) = (1/Pi)*Integral_{x=4..8} x^n*sqrt(8-x)/(2*sqrt(x-4));

a(n+1) = 4^n*F(-n,1/2;2;-1). (End)

From Gary W. Adamson, Jul 21 2011: (Start)

a(n) = upper left term of M^(n-1), M = an infinite square production matrix as follows:

5, 1, 0, 0, ...

1, 5, 1, 0, ...

1, 1, 5, 1, ...

1, 1, 1, 5, ...

... (End)

Recurrence: n*a(n) = 2*(6*n-7)*a(n-1) - 32*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

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

From Peter Bala, Feb 04 2024: (Start)

a(n+1) = Sum_{k = 0..n} binomial(n, k)* Catalan(k) * 4^(n-k).

a(n+1) = 4^n * hypergeom([-n, 1/2], [2], -1). (End)

Maple

seq(add(binomial(n-1, k)* (2*k)!/((k+1)*k!^2) * 4^(n-k-1), k = 0..n-1), n = 0..20); # Peter Bala, Feb 04 2024

Mathematica

CoefficientList[Series[1/2*(1-Sqrt[1-8*x]/Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

Prog

(PARI) x='x+O('x^66); concat([0], Vec((1-sqrt(1-8*x)/sqrt(1-4*x))/2 )) /* Joerg Arndt, Mar 31 2013 */

Crossrefs

Cf. A000108, A001653, A104497, A138240.

Sequence in context: A081187 A182401 A363308 * A045379 A053487 A277957

Adjacent sequences: A104495 A104496 A104497 * A104499 A104500 A104501

Keyword

easy,nonn

Author

Paul Barry, Mar 11 2005, Mar 07 2008

Status

approved