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Expansion of (1/2)*(1-sqrt(1-8*x)/sqrt(1-4*x)).
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%I #37 Feb 06 2024 11:29:59

%S 0,1,5,26,141,798,4706,28820,182461,1188406,7926102,53910828,

%T 372671250,2610977388,18498911268,132310178472,953981219997,

%U 6926326243110,50593306470542,371528742549692,2741187564459910,20310150708154564

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

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

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

%C a(n+1) is the fourth binomial transform of the Catalan numbers A000108. - _Paul Barry_, Oct 09 2010

%C 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

%H Vincenzo Librandi, <a href="/A104498/b104498.txt">Table of n, a(n) for n = 0..200</a>

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

%F From _Paul Barry_, Oct 09 2010: (Start)

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

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

%F From _Gary W. Adamson_, Jul 21 2011: (Start)

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

%F 5, 1, 0, 0, ...

%F 1, 5, 1, 0, ...

%F 1, 1, 5, 1, ...

%F 1, 1, 1, 5, ...

%F ... (End)

%F Recurrence: n*a(n) = 2*(6*n-7)*a(n-1) - 32*(n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012

%F a(n) ~ 2^(3*n-3/2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012

%F From _Peter Bala_, Feb 04 2024: (Start)

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

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

%p 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

%t CoefficientList[Series[1/2*(1-Sqrt[1-8*x]/Sqrt[1-4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)

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

%Y Cf. A000108, A001653, A104497, A138240.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Mar 11 2005, Mar 07 2008