A100087 Expansion of x/(sqrt(1-4*x^2) + x - 1).

1, 2, 4, 10, 24, 60, 148, 370, 920, 2300, 5736, 14340, 35808, 89520, 223668, 559170, 1397496, 3493740, 8732920, 21832300, 54575888, 136439720, 341082504, 852706260, 2131706864, 5329267160, 13322959888, 33307399720, 83267756400, 208169391000, 520420803060, 1301052007650

Offset: 0

Author

Paul Barry, Nov 03 2004

Keywords

• easy
• nonn

Comments

Inverse Chebyshev transform of (1-x^2)/((1-2*x)*(1+x^2)), the g.f. of A100088, under the mapping g(x) -> (1/sqrt(1-4*x^2))*g(x*c(x^2)) where c(x) is the g.f. of the Catalan numbers A000108. Equivalently, its image under the Chebyshev map A(x) -> ((1-x^2)/(1+x^2))*A(x/(1+x^2)) is A100088.

Transform of 1/(1-2*x) under the mapping g(x) -> g(x*c(x^2)). - Paul Barry, Jan 17 2005

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A100067, A100088.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( x/(Sqrt(1-4*x^2) +x-1) )); // G. C. Greubel, Jul 08 2022

Mathematica

CoefficientList[Series[x/(Sqrt[1-4*x^2]+x-1), {x, 0, 50}], x] (* Vaclav Kotesovec, Dec 06 2012 *)

PARI

my(x='x+O('x^66)); Vec(x/(sqrt(1-4*x^2)+x-1)) \\ Joerg Arndt, May 12 2013

SageMath

@CachedFunction
def A100067(n): return sum( binomial(n,k)*2^(n-2*k) for k in (0..(n//2)) )
def A100087(n): return (3/5)*A100067(n) + (1/5)*((1+(-1)^n) -2*I*(1-(-1)^n))*I^n*(-1)^floor(n/2)*binomial(n-1, floor(n/2))
[A100087(n) for n in (0..60)] # G. C. Greubel, Jul 08 2022

Formula

a(n) = Sum_{k=0..floor(n/2)} C(n, k)*(3*2^(n-2*k) + 2*cos(Pi*(n-2*k)/2) + 4*sin(Pi*(n-2*k)/2))/5.

a(n) = Sum_{k=0..floor(n/2)} C(n, k)*A100088(n-2*k).

a(n) = Sum_{k=0..n} k*C(n-1,(n-k)/2)*(1 + (-1)^(n-k))*2^k/(n+k). - Paul Barry, Jan 17 2005

D-finite with recurrence: 4*n*a(n) + 2*(2*n-7)*a(n-1) - (51*n-83)*a(n-2) - 8*(2*n-13)*a(n-3) + 140*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 22 2012

a(n) ~ 3*5^(n-1)/2^n. - Vaclav Kotesovec, Dec 06 2012