A158499 Expansion of (1 + sqrt(1-4*x))/(2-4*x).

1, 1, 1, 0, -5, -24, -90, -312, -1053, -3536, -11934, -40664, -140114, -488240, -1719380, -6113200, -21921245, -79200160, -288045110, -1053728920, -3874721030, -14313562480, -53093391980, -197669347600, -738398308850, -2766700765024

Offset: 0

Author

Paul Barry, Mar 20 2009

Keywords

• easy
• sign

Comments

Hankel transform is A056594 with g.f. 1/(1+x^2).

Row sums of the Riordan array (sqrt(1-4*x)/(1-2*x), x*c(x)^2), c(x) the g.f. of A000108.

The inverse Catalan transform yields A146559. - R. J. Mathar, Mar 20 2009

Links

• Matthew House, Table of n, a(n) for n = 0..1669
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Crossrefs

Cf. A000108, A056594, A146559, A158495.

Programs

Magma

[n le 2 select 1 else (6*(n-2)*Self(n-1) - 4*(2*n-5)*Self(n-2))/(n-1): n in [1..30]]; // G. C. Greubel, Mar 17 2025

Mathematica

CoefficientList[ Series[(1 + Sqrt[1 - 4x])/(2 - 4x), {x, 0, 26}], x] (* Robert G. Wilson v, Nov 08 2015 *)

PARI

my(x='x+O('x^33)); Vec(((1-4*x)+sqrt(1-4*x))/(2*(1-2*x)*sqrt(1-4*x))) \\ Altug Alkan, Nov 08 2015

SageMath

@CachedFunction
def a(n): # a = A158499
    if n<2: return 1
    else: return (6*(n-1)*a(n-1) - 4*(2*n-3)*a(n-2))/n
[a(n) for n in range(41)] # G. C. Greubel, Mar 17 2025

Formula

a(n) = Sum_{k=0..n} binomial(2*k,k)*A158495(n-k).

n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011

Extensions

Name edited by Matthew House, Nov 08 2015