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A203578
Exponential (or binomial) half-convolution of A000045 (Fibonacci) with itself.
2
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
0,3
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
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:
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.
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
Cf. A000045, 2*A014335 (exponential convolution), A032443.
Sequence in context: A059188 A080768 A370616 * A329442 A281486 A185895
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 13 2012
STATUS
approved