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Molecular topological indices of the triangular graphs.
2

%I #32 May 14 2022 03:54:33

%S 0,0,24,240,1080,3360,8400,18144,35280,63360,106920,171600,264264,

%T 393120,567840,799680,1101600,1488384,1976760,2585520,3335640,4250400,

%U 5355504,6679200,8252400,10108800,12285000,14820624,17758440,21144480

%N Molecular topological indices of the triangular graphs.

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

%H Reinhard Zumkeller, <a href="/A192849/b192849.txt">Table of n, a(n) for n = 1..10000</a>

%H G. D. Birkhoff, <a href="https://www.jstor.org/stable/1967597">A determinant formula for the number of ways of coloring a map</a>, Ann. Math., 14:42-4. See 2nd polynomial p. 5.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MolecularTopologicalIndex.html">Molecular Topological Index</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

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

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

%F G.f.: 24*x^3*(4*x+1)/(x-1)^6. - _Colin Barker_, Aug 07 2012

%F a(n) = A245334(n+1,4), n > 2. - _Reinhard Zumkeller_, Aug 31 2014

%F E.g.f.: x^3*(4 + 6*x + x^2)*exp(x). - _G. C. Greubel_, Jan 05 2019

%F From _Amiram Eldar_, May 14 2022: (Start)

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

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

%p [n*(n^2-1)*(n-2)^2$n=1..40]; # _Muniru A Asiru_, Jan 05 2019

%t Table[n*(n^2-1)*(n-2)^2, {n,1,40}] (* _G. C. Greubel_, Jan 05 2019 *)

%o (Haskell)

%o a192849 n = if n < 3 then 0 else a245334 (n + 1) 4

%o -- _Reinhard Zumkeller_, Aug 31 2014

%o (PARI) vector(40, n, n*(n^2 -1)*(n-2)^2) \\ _G. C. Greubel_, Jan 05 2019

%o (Magma) [n*(n^2 -1)*(n-2)^2: n in [1..40]]; // _G. C. Greubel_, Jan 05 2019

%o (Sage) [n*(n^2 -1)*(n-2)^2 for n in (1..40)] # _G. C. Greubel_, Jan 05 2019

%o (GAP) List([1..40], n -> n*(n^2 -1)*(n-2)^2); # _G. C. Greubel_, Jan 05 2019

%Y Cf. A027800, A245334.

%K nonn,easy

%O 1,3

%A _Eric W. Weisstein_, Jul 11 2011