A213820 Principal diagonal of the convolution array A213819.

2, 18, 60, 140, 270, 462, 728, 1080, 1530, 2090, 2772, 3588, 4550, 5670, 6960, 8432, 10098, 11970, 14060, 16380, 18942, 21758, 24840, 28200, 31850, 35802, 40068, 44660, 49590, 54870, 60512, 66528, 72930, 79730, 86940, 94572, 102638, 111150, 120120, 129560, 139482

Offset: 1

Author

Clark Kimberling, Jul 04 2012

Keywords

• nonn
• easy

Comments

Every term is even: a(n) = 2*A002414(n).

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

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002414, A213819, A328910.

Programs

Mathematica

(See A213819.)
a[n_] := 2*n^3 + n^2 - n; Array[a, 50] (* Amiram Eldar, Mar 12 2023 *)

Formula

a(n) = -n + n^2 + 2*n^3.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 5*x) and g(x) = (1-x)^4.

From Amiram Eldar, Mar 12 2023: (Start)

Sum_{n>=1} 1/a(n) = (4*log(2) - 1)/3.

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi - 4*log(2) + 1)/3. (End)