A203578 Exponential (or binomial) half-convolution of A000045 (Fibonacci) with itself.

0, 0, 2, 3, 14, 35, 155, 371, 1518, 3891, 15745, 40755, 161459, 426803, 1671175, 4469555, 17301630, 46805811, 179569163, 490156851, 1865624365, 5132989235, 19404565567, 53753361203, 201986220339, 562912506675, 2103942223775, 5894896300851, 21927151270703, 61732155503411

Offset: 0

Author

Wolfdieter Lang, Jan 13 2012

Keywords

• nonn
• easy

Comments

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see a comment on A203576 where also the rule for the e.g.f. is given.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1962
• Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.

Crossrefs

Cf. A000045, 2*A014335 (exponential convolution), A032443.

Programs

Mathematica

Table[Sum[Binomial[n,k]Fibonacci[k]Fibonacci[n-k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Mar 04 2013 *)

Formula

a(n) = sum(binomial(n,k)*F(k)*F(n-k),k=0..floor(n/2)), n>=0, with F(n)=A000045(n).

E.g.f.: (f(x)^2 + Fs2(x^2))/2, with the e.g.f. f(x) of A000045 and the o.g.f. Fs2(x):=sum((F(n)/n!)^2*x^n,n=0..infty) of the scaled squares. f(x)^2 = 2*exp(x)*(cosh((2*phi-1)*x)-1)/5 (see A000045 for f(x)) and Fs2(x^2) = (BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) - 2*BesselI(0,2*i*x))/5, with the golden section phi:=(1+sqrt(5))/2, the complex unit i, and for BesselI see Abramowitz-Stegun (reference and link given in A008277, p. 375, eq. 9.6.10). BesselI(0,2*sqrt(y)) = hypergeom([],[1],y) is the e.g.f. of the sequence {1/n!}.

Bisection:

a(2*k) = (A032443(k)*L(2*k) - (1 + (-1)^k*binomial(2*k,k)))/5 and a(2*k) = (2^(2*k)*L(2*k+1) - 1)/5, k>=0, with the Lucas numbers L(n)=A000032(n), and A032443(k)=(2^(2*k) + binomial(2*k,k))/2. - Wolfdieter Lang, Jan 16 2012.