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A213820 Principal diagonal of the convolution array A213819.

%I A213820 #39 Mar 12 2023 04:20:52
%S A213820 2,18,60,140,270,462,728,1080,1530,2090,2772,3588,4550,5670,6960,8432,
%T A213820 10098,11970,14060,16380,18942,21758,24840,28200,31850,35802,40068,
%U A213820 44660,49590,54870,60512,66528,72930,79730,86940,94572,102638,111150,120120,129560,139482
%N A213820 Principal diagonal of the convolution array A213819.
%C A213820 Every term is even: a(n) = 2*A002414(n).
%C A213820 a(n) is the first Zagreb index of the graph obtained by joining one vertex of a complete graph K[n] with each vertex of a second complete graph K[n]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.  Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - _Emeric Deutsch_, Nov 07 2016
%H A213820 Clark Kimberling, <a href="/A213820/b213820.txt">Table of n, a(n) for n = 1..1000</a>
%H A213820 Ivan Gutman and Kinkar C. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
%H A213820 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A213820 a(n) = -n + n^2 + 2*n^3.
%F A213820 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A213820 G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 5*x) and g(x) = (1-x)^4.
%F A213820 From _Amiram Eldar_, Mar 12 2023: (Start)
%F A213820 Sum_{n>=1} 1/a(n) = (4*log(2) - 1)/3.
%F A213820 Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi - 4*log(2) + 1)/3. (End)
%t A213820 (See A213819.)
%t A213820 a[n_] := 2*n^3 + n^2 - n; Array[a, 50] (* _Amiram Eldar_, Mar 12 2023 *)
%Y A213820 Cf. A002414, A213819, A328910.
%K A213820 nonn,easy
%O A213820 1,1
%A A213820 _Clark Kimberling_, Jul 04 2012