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A303212 Number of minimum total dominating sets in the n X n rook complement graph. 2

%I

%S 0,1,6,96,600,2400,7350,18816,42336,86400,163350,290400,490776,794976,

%T 1242150,1881600,2774400,3995136,5633766,7797600,10613400,14229600,

%U 18818646,24579456,31740000,40560000,51333750,64393056,80110296,98901600

%N Number of minimum total dominating sets in the n X n rook complement graph.

%C For n > 2, the minimum total dominating sets are any three vertices such that no two are in the same row or column. - _Andrew Howroyd_, Apr 20 2018

%H Colin Barker, <a href="/A303212/b303212.txt">Table of n, a(n) for n = 1..1000</a>

%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/TotalDominatingSet.html">Total Dominating Set</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) = A179058(n) for n > 2. - _Andrew Howroyd_, Apr 20 2018

%F From _Colin Barker_, Apr 20 2018: (Start)

%F G.f.: x^2*(1 - x + 75*x^2 + 19*x^3 + 41*x^4 - 21*x^5 + 7*x^6 - x^7) / (1 - x)^7.

%F a(n) = n^2*(2 - 3*n + n^2)^2 / 6 for n>2.

%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) for n>9.

%F (End)

%t Table[If[n == 2, 1, 6 Binomial[n, 3]^2], {n, 20}]

%t Join[{0, 1}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 6, 96, 600, 2400, 7350}, {3, 20}]]

%t CoefficientList[Series[x (-1 + x - 75 x^2 - 19 x^3 - 41 x^4 + 21 x^5 - 7 x^6 + x^7)/(-1 + x)^7, {x, 0, 20}], x]

%o (PARI) a(n) = if(n<3, n==2, 6*binomial(n,3)^2) \\ _Andrew Howroyd_, Apr 20 2018

%o (PARI) concat(0, Vec(x^2*(1 - x + 75*x^2 + 19*x^3 + 41*x^4 - 21*x^5 + 7*x^6 - x^7) / (1 - x)^7 + O(x^60))) \\ _Colin Barker_, Apr 20 2018

%Y Cf. A179058.

%K nonn,easy

%O 1,3

%A _Eric W. Weisstein_, Apr 19 2018

%E a(6)-a(30) from _Andrew Howroyd_, Apr 20 2018

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Last modified October 14 00:32 EDT 2019. Contains 327991 sequences. (Running on oeis4.)