OFFSET
0,3
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.
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:
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 13 2012
STATUS
approved