A127363 a(n) = Sum_{k=0..n} C(n,floor(k/2))*(-4)^(n-k).

1, -3, 14, -57, 246, -1038, 4424, -18777, 79846, -339258, 1442004, -6128202, 26045436, -110691948, 470442924, -1999378137, 8497365126, -36113785698, 153483619604, -652305322542, 2772297736276, -11782265148228, 50074627320864, -212817165231882, 904472953925596

Offset

0,2

Comments

Hankel transform is 5^n. In general, for r>=0, the sequence given by Sum_{k=0..n} C(n,floor(k/2))*(-r)^(n-k) has Hankel transform (r+1)^n. The sequence is the image of the sequence with g.f. (1+x)/(1+4x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.

Links

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

Formula

G.f.: (1/sqrt(1-4x^2))(1+x*c(x^2))/(1+4*x*c(x^2)).

Conjecture: 4*n*a(n) +(17*n-8)*a(n-1) +2*(-8*n-1)*a(n-2) +68*(-n+2)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ (-1)^n * 3 * 17^n / 4^(n+1). - Vaclav Kotesovec, Feb 12 2014

Mathematica

CoefficientList[Series[1/Sqrt[1-4*x^2] * (1+x*(1-Sqrt[1-4*x^2]) / (2*x^2)) / (1+4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

Crossrefs

Cf. A000108.

Sequence in context: A037093 A135926 A015523 * A133444 A126875 A110526

Adjacent sequences: A127360 A127361 A127362 * A127364 A127365 A127366

Keyword

easy,sign

Author

Paul Barry, Jan 11 2007

Extensions

More terms from Vincenzo Librandi, Feb 13 2014

Status

approved