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

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

0,5

Comments

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

Row sums of the Riordan array (sqrt(1-4x)/(1-2x),xc(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.

Formula

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

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

This conjecture has been proven. - Matthew House, Nov 08 2015

Mathematica

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

Prog

(PARI) 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

Crossrefs

Sequence in context: A089095 A220316 A220339 * A074085 A145914 A066316

Adjacent sequences: A158496 A158497 A158498 * A158500 A158501 A158502

Keyword

easy,sign

Author

Paul Barry, Mar 20 2009

Extensions

Name edited by Matthew House, Nov 08 2015

Status

approved