A337956 - OEIS

%I #13 Aug 12 2021 10:25:48

%S 1,15,126,730,3270,11991,37450,102726,253485,573265,1205556,2384460,

%T 4475926,8031765,13858860,23106196,37372545,58837851,90421570,

%U 135971430,200486286,290376955,413769126,580852650,804281725

%N Number of oriented colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

%C Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual.

%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) = binomial(binomial(n+1,2)+3,4) + binomial(binomial(n,2),4).

%F a(n) = n * (n+1) * (n^6 - n^5 + 7*n^4 + 29*n^3 + 16*n^2 - 4*n + 48) / 192.

%F a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 312*C(n,4) + 735*C(n,5) + 1020*C(n,6) + 735*C(n,7) + 210*C(n,8), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.

%F a(n) = A337957(n) + A234249(n+1) = 2*A337957(n) - A337958(n) = 2*A234249(n+1) + A337958(n).

%F From _Stefano Spezia_, Oct 04 2020: (Start)

%F G.f.: x*(1 + 6*x + 27*x^2 + 52*x^3 + 102*x^4 + 21*x^5 + x^6)/(1 - x)^9.

%F a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-8) for n > 8.

%F (End)

%t Table[Binomial[Binomial[n+1,2]+3,4] + Binomial[Binomial[n,2],4],{n,30}]

%Y Cf. A337957 (unoriented), A234249(n+1) (chiral), A337958 (achiral).

%Y Other elements: A331354 (hyperoctahedron edges, tesseract faces), A331358 (hyperoctahedron faces, tesseract edges), A337952 (hyperoctahedron facets, tesseract vertices).

%Y Other polychora: A337895 (5-cell), A338948 (24-cell), A338964 (120-cell, 600-cell).

%Y Row 4 of A325004 (orthotope facets, orthoplex vertices).

%K nonn,easy

%O 1,2

%A _Robert A. Russell_, Oct 03 2020