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A104498 Expansion of (1/2)(1-sqrt(1-8x)/sqrt(1-4x)). 7
0, 1, 5, 26, 141, 798, 4706, 28820, 182461, 1188406, 7926102, 53910828, 372671250, 2610977388, 18498911268, 132310178472, 953981219997, 6926326243110, 50593306470542, 371528742549692, 2741187564459910, 20310150708154564 (list; graph; refs; listen; history; text; internal format)
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

a(n+1) = (1/pi)*int(x^n*sqrt(8-x)/(2*sqrt(x-4)),x,4,8);

a(n+1) = 4^n*F(-n,1/2;2;-1). [From Paul Barry, Oct 09 2010]

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

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. A138240.

Sequence in context: A081911 A081187 A182401 * 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

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)