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

%I #21 Apr 16 2022 11:35:08

%S 0,24,576,4320,19200,63000,169344,395136,829440,1603800,2904000,

%T 4983264,8176896,12918360,19756800,29376000,42614784,60488856,

%U 84214080,115231200,155232000,206186904,270374016,350409600,449280000,570375000,717522624,895025376,1107697920

%N a(n) = 2*(n-1)^3*n^2*(n+1).

%C Molecular topological index of the n X n rook complement graph for n != 2.

%H G. C. Greubel, <a href="/A292282/b292282.txt">Table of n, a(n) for n = 1..5000</a>

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = 2*(n-1)^3*n^2*(n+1).

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

%F G.f.: -24*x^2*(1 + 17*x + 33*x^2 + 9*x^3)/(-1 + x)^7.

%F From _Amiram Eldar_, Apr 16 2022: (Start)

%F Sum_{n>=2} 1/a(n) = 51/32 - 3*Pi^2/16 + zeta(3)/4.

%F Sum_{n>=2} (-1)^n/a(n) = 2*log(2) + 3*zeta(3)/16 - 47/32 - Pi^2/96. (End)

%t Table[2 (-1 + n)^3 n^2 (1 + n), {n, 20}]

%t LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 24, 576, 4320, 19200, 63000, 169344}, 30]

%t CoefficientList[Series[-24 x (1 + 17 x + 33 x^2 + 9 x^3)/(-1 + x)^7, {x, 0, 20}], x]

%o (PARI) a(n)=2*(n-1)^3*n^2*(n+1) \\ _Charles R Greathouse IV_, Sep 14 2017

%o (Magma) [2*(n-1)^3*n^2*(n+1): n in [1..30]]; // _G. C. Greubel_, Dec 12 2017

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_, Sep 14 2017