|
%I #27 Dec 28 2020 15:49:08
%S 15,210,1365,5985,20475,58905,148995,341055,720720,1426425,2672670,
%T 4780230,8214570,13633830,21947850,34389810,52602165,78738660,
%U 115584315,166695375,236561325,330791175,456326325,621682425,837222750,1115465715,1471429260,1923014940
%N Number of ways to choose 4 points in an n X n X n triangular grid.
%C Sequence is column #5 of A084546: a(n) = A084546(n+1, 4).
%C All elements of the sequence are multiples of 15.
%C a(n-1) is the number of chiral pairs of colorings of the 8 cubic facets of a tesseract (hypercube) with Schläfli symbol {4,3,3} or of the 8 vertices of a hyperoctahedron with Schläfli symbol {3,3,4}. Both figures are regular 4-D polyhedra and they are mutually dual. Each member of a chiral pair is a reflection, but not a rotation, of the other. - _Robert A. Russell_, Oct 20 2020
%H Heinrich Ludwig, <a href="/A234249/b234249.txt">Table of n, a(n) for n = 3..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = n*(n + 1)*(n - 1)*(n + 2)*(n - 2)*(n + 3)*(n^2 + n - 4)/384.
%F a(n) = C(C(n + 1, 2), 4).
%F G.f.: -15*x^3*(x^2+5*x+1) / (x-1)^9. - _Colin Barker_, Feb 02 2014
%F From _Robert A. Russell_, Oct 20 2020: (Start)
%F a(n-1) = 15*C(n,4) + 135*C(n,5) + 330*C(n,6) + 315*C(n,7) + 105*C(n,8), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
%F a(n-1) = A337956(n) - A337957(n) = (A337956(n) - A337958(n)) / 2 = A337957(n) - A337958(n).
%F a(n-1) = A325006(4,n). (End)
%p A234249:=n->n*(n + 1)*(n - 1)*(n + 2)*(n - 2)*(n + 3)*(n^2 + n - 4)/384: seq(A234249(n), n=3..40); # _Wesley Ivan Hurt_, Jan 10 2017
%t Table[Binomial[Binomial[n,2],4], {n,4,30}] (* _Robert A. Russell_, Oct 20 2020 *)
%o (PARI) Vec(-15*x^3*(x^2+5*x+1)/(x-1)^9 + O(x^100)) \\ _Colin Barker_, Feb 02 2014
%Y Cf. A084546, A050534 (number of ways to choose 2 points), A093566 (3 points), A231653.
%Y Cf. A337956 (oriented), A337956 (unoriented), A337956 (achiral) colorings, A331356 (hyperoctahedron edges, tesseract faces), A331360 (hyperoctahedron faces, tesseract edges), A337954 (hyperoctahedron facets, tesseract vertices).
%Y Other polychora: A000389 (5-cell), A338950 (24-cell), A338966 (120-cell, 600-cell).
%Y Row 4 of A325006 (orthotope facets, orthoplex vertices).
%K nonn,easy
%O 3,1
%A _Heinrich Ludwig_, Feb 02 2014
|
