0,3

G.f. satisfies: A(x) = x/(series reversion of x*G098614(x)), where G098614 is the g.f. for A098614 = {1*1, 1*1, 2*2, 3*5, 5*14, 8*42, 13*132, ...}.

Hankel transform is 2^n. Image of F(n+1) under the Riordan array (c(x^2),xc(x^2)), c(x) the g.f. of A000108. The sequence 0,1,1,3,5,... has general term sum{k=0..floor(n/2), (C(n-1,k)-C(n-1,k-1))F(n-2k)}. It is the image of the Fibonacci numbers under the transform of generating functions g(x)-> g(xc(x^2)), c(x) the g.f. of A000108. This sequence has Hankel transform -(-4)^((n-1)/2)(1-(-1)^n)/2. - Paul Barry (pbarry(AT)wit.ie), Oct 01 2007

The sequence of fractions 1, 1/2, 3/4, 5/8, 13/16, 25/32, ... or a(n)/2^n is the image of F(n+1) under the Chebyshev related (rational) Riordan array c((x/2)^2),(x/2)c((x/2)^2)) where c(x) is the g.f. of A000108. The Hankel transform of this fraction sequence is 1/(2^(n^2)). - Paul Barry (pbarry(AT)wit.ie), Jun 17 2008

G.f.: A(x) = (sqrt(1-4*x^2) + x)/(1-5*x^2). a(2*n) = A046748(n); a(2*n+1) = 5^n.

a(n)=sum{k=0..floor((n+1)/2), (C(n,k)-C(n,k-1))*F(n-2k+1)}; - Paul Barry (pbarry(AT)wit.ie), Oct 01 2007

(PARI) a(n)=polcoeff((sqrt(1-4*x^2+x^2*O(x^n))+x)/(1-5*x^2), n)

Cf. A098614, A046748.

Sequence in context: A026709 A082010 A110494 this_sequence A026720 A026003 A103792

Adjacent sequences: A098612 A098613 A098614 this_sequence A098616 A098617 A098618

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 14 2004