A141343 -id:A141343 - cp's OEIS frontend

A141342 A transform of the Fibonacci numbers.

1, 1, -1, 3, -13, 65, -353, 2025, -12077, 74143, -465481, 2974863, -19289821, 126594191, -839273105, 5612483619, -37814455781, 256447068841, -1749182184793, 11991887667273, -82588248514885, 571118483653841

Offset: 0

Paul Barry, Jun 26 2008

A transform of F(n+1) by the inverse of the Riordan array (1, x*(1+x)/(1-2*x)).
Equivalently, row sums of the inverse of the Riordan array (1, x/(2-sqrt(1+4*x)).
Hankel transform is alternating sign version of A083667.

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

Cf. A141343.

CoefficientList[Series[1/(1-2*x-2*x^2+x*Sqrt[1+8*x+4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

x='x+O('x^50); Vec(1/(1-2*x-2*x^2+x*sqrt(1+8*x+4*x^2))) \\ G. C. Greubel, Mar 21 2017

G.f.: 1/(1-2*x-2*x^2+x*sqrt(1+8*x+4*x^2)).
Conjecture: (n-1)*a(n) +4*(n-4)*a(n-1) + (65-29*n)*a(n-2) +12*(7-2*n)*a(n-3)+ 4*(4-n)*a(n-4) =0. - R. J. Mathar, Nov 14 2011
a(n) ~ (-1)^n * (5*sqrt(3)-14) * sqrt(2*sqrt(3)-3) * 2^(n+1/2) * (2+sqrt(3))^n / (121 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014

A141344 Expansion of (2-sqrt(1+4x))/(2-x-sqrt(1+4x)).

Original entry on oeis.org

1, 1, 3, 7, 19, 45, 123, 285, 807, 1771, 5407, 10587, 37627, 57619, 279783, 231615, 2307339, -387531, 21769251, -28249347, 235837791, -539858235, 2857845723, -8509970007, 37342507167, -126289733319, 510715973643, -1837291760147

Offset: 0

Paul Barry, Jun 26 2008

Row sums of A141343. Hankel transform is 2^n.

Image of A052961 under the Riordan array (c(-x),xc(-x)^2), c(x) the g.f. of A000108. [From Paul Barry, Jan 29 2009]

CoefficientList[Series[(2-Sqrt[1+4x])/(2-x-Sqrt[1+4x]),{x,0,30}],x] (* Harvey P. Dale, Jan 14 2013 *)

Conjectured to be D-finite with recurrence: 3*(n-1)*a(n) +2*(2*n-11)*a(n-1) +(79-31*n)*a(n-2) +2*(2*n-5)*a(n-3)=0. - R. J. Mathar, Oct 25 2012

cp's OEIS Frontend

A141342 A transform of the Fibonacci numbers.

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