A100066 Expansion of x/((1-x)*sqrt(1-4*x^2)).

0, 1, 1, 3, 3, 9, 9, 29, 29, 99, 99, 351, 351, 1275, 1275, 4707, 4707, 17577, 17577, 66197, 66197, 250953, 250953, 956385, 956385, 3660541, 3660541, 14061141, 14061141, 54177741, 54177741, 209295261, 209295261, 810375651, 810375651

Offset: 0

Author

Paul Barry, Nov 02 2004

Keywords

• nonn
• easy

Comments

An inverse Chebyshev transform of x/(1-x+x^2), where the Chebyshev transform of g(x) is ((1-x^2)/(1+x^2))*g(x/(1+x^2)) and the inverse transform maps a g.f. A(x) to (1/sqrt(1-4*x^2))*A(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108.

Hankel transform of a(n+1) is A120582. The Hankel transform of a(n) is (-1)*[x^n] x/(1+2*x-4*x^2). - Paul Barry, Mar 29 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi).

Crossrefs

Cf. A000108, A006134, A120582.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
[0] cat Coefficients(R!( x/((1-x)*Sqrt(1-4*x^2)) )); // G. C. Greubel, Mar 17 2025

Maple

a:=n->sum(binomial(2*j,j), j=0..n): seq(a(n/2), n=-1..40); # Zerinvary Lajos, Apr 30 2007

Mathematica

CoefficientList[Series[x/((1-x)*Sqrt[1-4*x^2]), {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

PARI

my(x='x+O('x^41)); concat(0, Vec(x/((1-x)*sqrt(1-4*x^2)))) \\ G. C. Greubel, Jan 30 2017; Mar 17 2025

SageMath

def b(n): return sum(binomial(2*k,k) for k in range(n+1))
def A100066(n): return b((n-1)//2)
print([A100066(n) for n in range(41)]) # G. C. Greubel, Mar 17 2025

Formula

a(n) = Sum_{k=0..n} if(mod(n-k, 2)=0, binomial(n, (n-k)/2) * 2*sin(Pi*k/3) / sqrt(3)).

a(n) = Sum_{k=0..n} binomial(n, (n-k)/2)*(1+(-1)^(n-k))*sin(Pi*k/3)/sqrt(3).

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

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

a(2n-1) = a(2n) = A006134(n-1) = Sum_{k=0..n}( (2*k)!/(k!)^2 ) for n > 0. - Alexander Adamchuk, Feb 23 2007

From Paul Barry, Mar 29 2010: (Start)

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

E.g.f.: Integral_{t=0..x} exp(x-t)*Bessel_I(0,2t). (End)

D-finite with recurrence: (n-1)*a(n) - (n-1)*a(n-1) - 4*(n-2)*a(n-2) + 4*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 24 2012

a(n) ~ (3-(-1)^n) * 2^(n+1/2) / (6*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 12 2014