cp's OEIS Frontend

A203579 Exponential (or binomial) convolution of A000032 (Lucas) with itself, divided by 2.

%I A203579 #15 Dec 28 2020 20:33:22
%S A203579 2,2,7,17,57,177,577,1857,6017,19457,62977,203777,659457,2134017,
%T A203579 6905857,22347777,72318977,234029057,757334017,2450784257,7930904577,
%U A203579 25664946177,83053510657,268766806017,869747654657,2814562533377,9108115685377,29474481504257
%N A203579 Exponential (or binomial) convolution of A000032 (Lucas) with itself, divided by 2.
%H A203579 Michael De Vlieger, <a href="/A203579/b203579.txt">Table of n, a(n) for n = 0..1961</a>
%H A203579 Sergio Falcon, <a href="https://doi.org/10.7546/nntdm.2020.26.3.96-106">Half self-convolution of the k-Fibonacci sequence</a>, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
%F A203579 a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..n)/2, n>=0, with L(n)=A000032(n).
%F A203579 E.g.f.: (1/2)*(exp(phi*x)+exp(-(phi-1)*x))^2 =
%F A203579   exp(x)*(cosh(sqrt(5)*x)+1), with the golden section phi:=(1+sqrt(5))/2. (See the e.g.f. of A000032).
%F A203579 a(n) = 2^(n-1)*L(n) + 1.
%F A203579 a(n) = 5*A014335(n) + 2. - _Vladimir Reshetnikov_, Oct 06 2016
%e A203579 With A000032 = {2,1,3,4,7,...},
%e A203579   2*a(4) = 1*2*7 + 4*1*4 + 6*3*3 + 4*4*1 + 1*7*2 = 114.
%t A203579 Array[Sum[Binomial[#, k] LucasL[k] LucasL[# - k], {k, 0, #}]/2 &, 28, 0] (* _Michael De Vlieger_, Dec 28 2020 *)
%Y A203579 Cf. A000032, A014335.
%K A203579 nonn,easy
%O A203579 0,1
%A A203579 _Wolfdieter Lang_, Jan 14 2012