A156906 Transform of Fibonacci(n+1) with Hankel transform (-1)^binomial(n+1,2) * Fibonacci(n+1).

1, 0, 1, -1, 0, 0, 2, -2, -3, 3, 11, -11, -31, 31, 101, -101, -328, 328, 1102, -1102, -3760, 3760, 13036, -13036, -45750, 45750, 162262, -162262, -580638, 580638, 2093802, -2093802, -7601043, 7601043, 27756627, -27756627, -101888163

Offset: 0

Author

Paul Barry, Feb 17 2009

Keywords

• easy
• sign

Comments

Hankel transform is (-1)^binomial(n+1,2) * Fibonacci(n+1).

Image of Fibonacci(n+1) by the Riordan array (1/(1 + x*c(-x^2)), x*c(-x^2)/(1 + x*c(-x^2))) = (1/(1-x), x*(1-x)/(1-2*x))^{-1} = A055587^{-1}.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000045, A000108, A055587.

Programs

Maple

cx := (1-sqrt(1-4*x))/2/x ;
A156906 := proc(n)
        1+x^2*subs(x=-x^2,cx)/(1+x) ;
        coeftayl(%,x=0,n) ;
end proc:
seq(A156906(n), n=0..40);  # R. J. Mathar, Jul 28 2016

Mathematica

a[n_]:= (1/2)*(2*Boole[n==0] -(-1)^n + Sum[(-1)^(n+j+1)*Binomial[2*j, j]/(2*j-1), {j, 0, Floor[n/2]}]); Table[a[n], {n, 0, 60}] (* G. C. Greubel, Jun 14 2021 *)

Sage

def A156906(n): return (1/2)*( 2*bool(n==0) - (-1)^n + sum( (-1)^(n+j+1)*binomial( 2*j, j)/(2*j-1) for j in (0..n//2)) )
[A156906(n) for n in (0..40)] # G. C. Greubel, Jun 14 2021

Formula

G.f.: (1 + 2*x + sqrt(1+4*x^2))/(2*(1+x)) = 1 + x^2*c(-x^2)/(1+x) where c(x) is the g.f. of A000108;

G.f.: 1/(1 - x^2/(1 + x + 2*x^2/(1 - x/2 + 3*x^2/4/(1 + x/6 + 10*x^2/9/(1 - x/15 + 24/25*x^2/(1 + ...)))))) (continued fraction);

In the continued fraction expansion of the g.f. the general term is 1 - x*if(n=0, 0, (-1)^n/(Fibonacci(n)*Fibonacci(n+1))) + x^2*(-0^n + Fibonacci(n)*Fibonacci(n+2) )/Fibonacci(n+1)^2.

a(n) = Sum_{k=0..n} (-1)^(n-k)*b(k), where b(n) = binomial(1,n) -A000108((n-2)/2) * (-1)^(n/2) * (1+(-1)^n)/2 and b(0) = 1.

n*a(n) = -n*a(n-1) -4*(n-3)*a(n-2) -4*(n-3)*a(n-3). - R. J. Mathar, Nov 14 2011

a(n) = (1/2)*( 2*[n=0] - (-1)^n + Sum_{j=0..floor(n/2)} (-1)^(n+j+1)*binomial(2*j, j)/(2*j-1) ). - G. C. Greubel, Jun 14 2021