login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A126966 Expansion of sqrt(1 - 4*x)/(1 - 2*x). 5
1, 0, -2, -8, -26, -80, -244, -752, -2362, -7584, -24892, -83376, -284324, -984672, -3455144, -12259168, -43908026, -158531392, -576352364, -2107982128, -7750490636, -28629222112, -106190978264, -395347083808, -1476813394916, -5533435084480, -20790762971864, -78316232088032 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Hankel transform is 2^n*(-1)^binomial(n+1, 2) = A120617(n). - Paul Barry, Feb 08 2008

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = -Sum_{j=0..n} ( 2^j*binomial(2n-2j, n-j)/(2n-2j-1) ). - Emeric Deutsch, Mar 25 2007

D-finite with recurrence: n*a(n) + 6*(1-n)*a(n-1) + 4*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011, corrected Feb 17 2020

a(n) ~ -4^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013

a(n) = 2^n*i + CatalanNumber(n)*hypergeom([1, n + 1/2], [n + 2], 2). - Peter Luschny, Aug 04 2020

MAPLE

a := n -> -add(2^j*binomial(2*n-2*j, n-j)/(2*n-2*j-1), j=0..n):

seq(a(n), n=0..30); # Emeric Deutsch, Mar 25 2007

# second Maple program:

CatalanNumber := n -> binomial(2*n, n)/(n+1):

a := n -> 2^n*I + CatalanNumber(n)*simplify(hypergeom([1, n + 1/2], [n + 2], 2)):

seq(a(n), n=0..26); # Peter Luschny, Aug 04 2020

MATHEMATICA

CoefficientList[Series[Sqrt[1-4*x]/(1-2*x), {x, 0, 30}], x] (* G. C. Greubel, Jan 31 2017 *)

PROG

(PARI) Vec(sqrt(1-4*x)/(1-2*x) + O(x^30)) \\ G. C. Greubel, Jan 31 2017

(MAGMA) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt(1-4*x)/(1-2*x) )); // G. C. Greubel, Jan 29 2020

(Sage)

def A126966_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( sqrt(1-4*x)/(1-2*x) ).list()

A126966_list(30) # G. C. Greubel, Jan 29 2020

(GAP) List([0..30], n-> (-1)*Sum([0..n], j-> 2^j*Binomial(2*(n-j), n-j)/(2*(n-j) -1) )); # G. C. Greubel, Jan 29 2020

CROSSREFS

Cf. A000108, A126967.

Sequence in context: A103453 A024023 A295137 * A002930 A060410 A128634

Adjacent sequences:  A126963 A126964 A126965 * A126967 A126968 A126969

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Mar 22 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 23 13:59 EDT 2021. Contains 345402 sequences. (Running on oeis4.)