A181641 Expansion of sqrt(1-4*x)/(1+x).

1, -3, 1, -5, -5, -23, -61, -203, -655, -2205, -7519, -26073, -91499, -324525, -1161275, -4187605, -15202085, -55513255, -203776325, -751501075, -2783025305, -10345215535, -38587318505, -144377808775, -541741418525

Offset

0,2

Comments

Hankel transform is alternating sign version of A082762.

References

Joseph Edwards, Differential Calculus with Applications and Numerous Examples: An Elementary Treatise, 1886.

Links

Table of n, a(n) for n=0..24.

Formula

G.f.: sqrt(1-4*x)/(1+x).

a(n) = Sum_{k=0..n} (-1)^(n-k)*C(2k,k)/(1-2k).

D-finite with recurrence: n*a(n) + 3*(2-n)*a(n-1) + 2*(3-2*n)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011.

G.f. A(x) = 1/G(0), where G(k) = 1 + x/(1 - (4*k+2)/((4*k+2) + (k+1)/G(k+1))); (continued fraction 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 24 2012

G.f.: 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013

a(n) = binomial(2*n,n) * hypergeom([1, -n], [1/2], 5/4). - Vladimir Reshetnikov, Oct 02 2016

a(n) ~ -2^(2*n+1) / (5*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 03 2016

Mathematica

CoefficientList[Series[Sqrt[1-4x]/(1+x), {x, 0, 30}], x] (* Harvey P. Dale, Jul 20 2011 *)
Table[FullSimplify[(-1)^n*Sqrt[5] - 1/(1+2*n)*Binomial[2*(1+n), 1+n] * Hypergeometric2F1[1, 1/2+n, 2+n, -4]], {n, 0, 20}] (* Vaclav Kotesovec, Jan 31 2014 *)
Table[Binomial[2 n, n] Hypergeometric2F1[1, -n, 1/2, 5/4], {n, 0, 30}] (* Vladimir Reshetnikov, Oct 02 2016 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k, k)/(1-2*k)); \\ Michel Marcus, Oct 03 2016

Crossrefs

Cf. A082762.

Sequence in context: A220479 A146913 A146252 * A049266 A089028 A209758

Adjacent sequences: A181638 A181639 A181640 * A181642 A181643 A181644

Keyword

easy,sign

Author

Paul Barry, Nov 03 2010

Status

approved