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%I #14 Sep 25 2019 05:48:54
%S 4,2,8,17,84,177,737,1857,7732,19457,78223,203777,809145,2134017,
%T 8349013,22347777,86533892,234029057,897748577,2450784257,9328491339,
%U 25664946177,97021416973,268766806017,1009936510009,2814562533377
%N Exponential (or binomial) half-convolution of A000032 (Lucas) with itself.
%C 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.
%C The other half of this exponential half-convolution is found under A204450.
%F a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..floor(n/2)), n>=0, with L(n)=A000032(n).
%F 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}.
%F l(x)^2 = 2*exp(x)*(cosh(sqrt(5)*x)+1) (see 2*A203579).
%F 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.
%F BesselI(0,2*sqrt(x)) = hypergeom([],[1],x) is the e.g.f. of {1/n!}.
%F 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).
%e With A000032 = {2, 1, 3, 4, 7, 11,...}
%e a(4) = 1*2*7 + 4*1*4 + 6*3*3 = 84,
%e a(5) = 1*2*11 + 5*1*7 + 10*3*4 = 177.
%t Table[Sum[Binomial[n, k]*LucasL[k]*LucasL[n-k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* _Vaclav Kotesovec_, Sep 25 2019 *)
%Y Cf. A000032, 2*A203579 (exponential convolution), A204450.
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Jan 16 2012
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