A204449 Exponential (or binomial) half-convolution of A000032 (Lucas) with itself.

4, 2, 8, 17, 84, 177, 737, 1857, 7732, 19457, 78223, 203777, 809145, 2134017, 8349013, 22347777, 86533892, 234029057, 897748577, 2450784257, 9328491339, 25664946177, 97021416973, 268766806017, 1009936510009, 2814562533377

Offset: 0

Author

Wolfdieter Lang, Jan 16 2012

Keywords

• nonn
• easy

Comments

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

The other half of this exponential half-convolution is found under A204450.

Examples

			With A000032 = {2, 1, 3, 4, 7, 11,...}
  a(4) = 1*2*7 + 4*1*4 + 6*3*3 = 84,
  a(5) = 1*2*11 + 5*1*7 + 10*3*4 = 177.
		

Crossrefs

Cf. A000032, 2*A203579 (exponential convolution), A204450.

Programs

Mathematica

Table[Sum[Binomial[n, k]*LucasL[k]*LucasL[n-k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 25 2019 *)

Formula

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..floor(n/2)), n>=0, with L(n)=A000032(n).

E.g.f.: (l(x)^2 + L2(x^2))/2 with the e.g.f. l(x) of A000032, and the o.g.f. L2(x) of the sequence {(L(n)/n!)^2}.

l(x)^2 = 2*exp(x)*(cosh(sqrt(5)*x)+1) (see 2*A203579).

L2(x^2) = BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) + 2*BesselI(0,2*I*x), with the golden section phi:=(1+sqrt(5))/2, and for BesselI see Abramowitz-Stegun (reference and link given under A008277), p. 375, eq. 9.6.10.

BesselI(0,2*sqrt(x)) = hypergeom([],[1],x) is the e.g.f. of {1/n!}.

Bisection: a(2*k) = (2^(2*k)+binomial(2*k,k))*L(2*k)/2 +1 + ((-1)^k)*binomial(2*k,k), a(2*k+1) = 2^(2*k)*L(2*k+1)+1, k>=0. For (2^(2*k)+binomial(2*k,k))/2 see A032443(k).