A192849 Molecular topological indices of the triangular graphs.

0, 0, 24, 240, 1080, 3360, 8400, 18144, 35280, 63360, 106920, 171600, 264264, 393120, 567840, 799680, 1101600, 1488384, 1976760, 2585520, 3335640, 4250400, 5355504, 6679200, 8252400, 10108800, 12285000, 14820624, 17758440, 21144480

Offset: 1

Author

Eric W. Weisstein, Jul 11 2011

Keywords

• nonn
• easy

Comments

Triangular graphs are defined for n>=2; extended to n=1 using closed form.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• G. D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. Math., 14:42-4. See 2nd polynomial p. 5.
• Eric Weisstein's World of Mathematics, Molecular Topological Index.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A027800, A245334.

Programs

GAP

List([1..40], n -> n*(n^2 -1)*(n-2)^2); # G. C. Greubel, Jan 05 2019

Haskell

a192849 n = if n < 3 then 0 else a245334 (n + 1) 4
-- Reinhard Zumkeller, Aug 31 2014

Magma

[n*(n^2 -1)*(n-2)^2: n in [1..40]]; // G. C. Greubel, Jan 05 2019

Maple

[n*(n^2-1)*(n-2)^2$n=1..40]; # Muniru A Asiru, Jan 05 2019

Mathematica

Table[n*(n^2-1)*(n-2)^2, {n,1,40}] (* G. C. Greubel, Jan 05 2019 *)

PARI

vector(40, n, n*(n^2 -1)*(n-2)^2) \\ G. C. Greubel, Jan 05 2019

Sage

[n*(n^2 -1)*(n-2)^2 for n in (1..40)] # G. C. Greubel, Jan 05 2019

Formula

a(n) = n*(n^2 - 1)*(n-2)^2.

a(n) = 24*A027800(n-3).

G.f.: 24*x^3*(4*x+1)/(x-1)^6. - Colin Barker, Aug 07 2012

a(n) = A245334(n+1,4), n > 2. - Reinhard Zumkeller, Aug 31 2014

E.g.f.: x^3*(4 + 6*x + x^2)*exp(x). - G. C. Greubel, Jan 05 2019

From Amiram Eldar, May 14 2022: (Start)

Sum_{n>=3} 1/a(n) = Pi^2/36 - 49/216.

Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/72 - 10*log(2)/9 + 145/216. (End)