A126079 G.f.: (1-2*x)*sqrt(1-4*x).

1, -4, 2, 0, -2, -8, -28, -96, -330, -1144, -4004, -14144, -50388, -180880, -653752, -2377280, -8691930, -31935960, -117858900, -436698240, -1623971580, -6059188080, -22676052360, -85100059200, -320188972740, -1207569840048, -4564276213608, -17286920538496, -65597689543400

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

From Emeric Deutsch, Mar 25 2007: (Start)

a(n) = binomial(2n,n) - 6*binomial(2n-2,n-1) + 8*binomial(2n-4,n-2).

a(n) = (3-n)*binomial(2n,n)/((2*n-3)*(2*n-1)). (End)

E.g.f.: 1 - 4*x + 2*x^2 - x^2*Q(0), where Q(k)= 1 - 2*x/(k+3 - (k+3)*(2*k+3)/(2*k+3 - (k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013

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

D-finite with recurrence: n*a(n) +2*(-3*n+5)*a(n-1) +4*(2*n-7)*a(n-2)=0. - R. J. Mathar, Jan 23 2020

Maple

a:=n->(3-n)*binomial(2*n, n)/(2*n-3)/(2*n-1): seq(a(n), n=0..30); # Emeric Deutsch, Mar 25 2007
A126079List := proc(m) local A, P, n; A := [1, -4, 2, 0]; P := [-2, 0];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A126079List(27); # Peter Luschny, Mar 26 2022

Mathematica

CoefficientList[Series[(1-2x)Sqrt[1-4x], {x, 0, 40}], x] (* Harvey P. Dale, Mar 28 2014 *)

Prog

(Magma) [(3-n)*Binomial(2*n, n)/((2*n-3)*(2*n-1)): n in [0..30]]; // Vincenzo Librandi, Mar 29 2014

Crossrefs

Cf. A126966, A126089.

Sequence in context: A297331 A028956 A129681 * A106220 A176066 A309987

Adjacent sequences: A126076 A126077 A126078 * A126080 A126081 A126082

Keyword

sign

Author

N. J. A. Sloane, Mar 22 2007

Status

approved