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A204449 Exponential (or binomial) half-convolution of A000032 (Lucas) with itself. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 29 08:26 EST 2020. Contains 332355 sequences. (Running on oeis4.)