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A289228 Number of ways to select 7 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.

%I A289228 #8 Jul 01 2017 08:26:12
%S A289228 0,2910,404746,12025068,165279612,1405048082,8605979390,41555851716,
%T A289228 167529980320,586136559350,1829074790082,5193890370940,13625393372916,
%U A289228 33410188057962,77284672892438,169909353488372,357177283295160,721559475338446,1406717921047994,2656028041092876
%N A289228 Number of ways to select 7 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.
%C A289228 Rotations and reflections of a selection are regarded as different.
%H A289228 Heinrich Ludwig, <a href="/A289228/b289228.txt">Table of n, a(n) for n = 6..100</a>
%H A289228 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
%F A289228 a(n) = (n^14 -14*n^13 -182*n^12 +3416*n^11 +9072*n^10 -342062*n^9 +296688*n^8 +17893944*n^7 -48845153*n^6 -511039228*n^5 +2041220174*n^4 +7429535400*n^3 -37737333320*n^2 -41483946096*n +262680697440)/5040 for n>=7.
%F A289228 G.f.: 2*x^7*(1455 + 180548*x + 3129714*x^2 + 13038936*x^3 + 4149381*x^4 - 21524480*x^5 + 3658074*x^6 + 12791138*x^7 - 6864973*x^8 - 1299402*x^9 + 1667400*x^10 - 272962*x^11 + 37953*x^12 - 60178*x^13 + 16036*x^14) / (1 - x)^15. - _Colin Barker_, Jul 01 2017
%e A289228 There are 2910 ways to choose seven 2 X 2 X 2 triangles (aaa, ..., ggg) from a 7 X 7 X 7 point grid, for example:
%e A289228           a
%e A289228          a a
%e A289228         b . c
%e A289228        b b c c
%e A289228       d d . e e
%e A289228      f d . . e g
%e A289228     f f . . . g g
%e A289228 Note: aaa, ..., ggg are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
%o A289228 (PARI) concat(0, Vec(2*x^7*(1455 + 180548*x + 3129714*x^2 + 13038936*x^3 + 4149381*x^4 - 21524480*x^5 + 3658074*x^6 + 12791138*x^7 - 6864973*x^8 - 1299402*x^9 + 1667400*x^10 - 272962*x^11 + 37953*x^12 - 60178*x^13 + 16036*x^14) / (1 - x)^15 + O(x^40))) \\ _Colin Barker_, Jul 01 2017
%Y A289228 Cf. A289222, A289223, A289224, A289225, A289226, A289227.
%K A289228 nonn,easy
%O A289228 6,2
%A A289228 _Heinrich Ludwig_, Jul 01 2017