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

0,1

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.

Links

Table of n, a(n) for n=0..25.

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).

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A110622 A130078 A230900 * A172393 A245340 A040174

Adjacent sequences: A204446 A204447 A204448 * A204450 A204451 A204452

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 16 2012

Status

approved