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%I #17 Dec 28 2020 20:33:16
%S 0,0,2,3,14,35,155,371,1518,3891,15745,40755,161459,426803,1671175,
%T 4469555,17301630,46805811,179569163,490156851,1865624365,5132989235,
%U 19404565567,53753361203,201986220339,562912506675,2103942223775,5894896300851,21927151270703,61732155503411
%N Exponential (or binomial) half-convolution of A000045 (Fibonacci) with itself.
%C 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.
%H Michael De Vlieger, <a href="/A203578/b203578.txt">Table of n, a(n) for n = 0..1962</a>
%H Sergio Falcon, <a href="https://doi.org/10.7546/nntdm.2020.26.3.96-106">Half self-convolution of the k-Fibonacci sequence</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
%F a(n) = sum(binomial(n,k)*F(k)*F(n-k),k=0..floor(n/2)), n>=0, with F(n)=A000045(n).
%F 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!}.
%F Bisection:
%F 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.
%t Table[Sum[Binomial[n,k]Fibonacci[k]Fibonacci[n-k],{k,0,Floor[n/2]}],{n,0,30}] (* _Harvey P. Dale_, Mar 04 2013 *)
%Y Cf. A000045, 2*A014335 (exponential convolution), A032443.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Jan 13 2012
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