A292058 - OEIS

%I #16 Apr 16 2022 11:35:00

%S 0,10,54,168,400,810,1470,2464,3888,5850,8470,11880,16224,21658,28350,

%T 36480,46240,57834,71478,87400,105840,127050,151294,178848,210000,

%U 245050,284310,328104,376768,430650,490110,555520,627264,705738,791350,884520,985680,1095274

%N Wiener index of the n X n rook complement graph.

%C Sequence extended to a(2) using the formula/recurrence (the graph is disconnected for n = 2).

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

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

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) -5 *a(n-4) + a(n-5).

%F G.f.: 2*x^2*(-5 - 2*x + x^2)/(-1 + x)^5.

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

%F Sum_{n>=2} 1/a(n) = 265/216 - Pi^2/9.

%F Sum_{n>=2} (-1)^n/a(n) = Pi^2/18 + 8*log(2)/9 - 233/216. (End)

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

%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 10, 54, 168, 400}, 20]

%t CoefficientList[Series[2 x (-5 - 2 x + x^2)/(-1 + x)^5, {x, 0, 20}], x]

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_, Sep 08 2017