A338964 -id:A338964 - cp's OEIS frontend

A337956 Number of oriented colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

1, 15, 126, 730, 3270, 11991, 37450, 102726, 253485, 573265, 1205556, 2384460, 4475926, 8031765, 13858860, 23106196, 37372545, 58837851, 90421570, 135971430, 200486286, 290376955, 413769126, 580852650, 804281725

Offset: 1

Robert A. Russell, Oct 03 2020

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.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A337957 (unoriented), A234249(n+1) (chiral), A337958 (achiral).
Other elements: A331354 (hyperoctahedron edges, tesseract faces), A331358 (hyperoctahedron faces, tesseract edges), A337952 (hyperoctahedron facets, tesseract vertices).
Other polychora: A337895 (5-cell), A338948 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A325004 (orthotope facets, orthoplex vertices).

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

a(n) = binomial(binomial(n+1,2)+3,4) + binomial(binomial(n,2),4).
a(n) = n * (n+1) * (n^6 - n^5 + 7*n^4 + 29*n^3 + 16*n^2 - 4*n + 48) / 192.
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.
a(n) = A337957(n) + A234249(n+1) = 2*A337957(n) - A337958(n) = 2*A234249(n+1) + A337958(n).
From Stefano Spezia, Oct 04 2020: (Start)
G.f.: x*(1 + 6*x + 27*x^2 + 52*x^3 + 102*x^4 + 21*x^5 + x^6)/(1 - x)^9.
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.
(End)

A338980 Number of oriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using exactly n colors.

Original entry on oeis.org

0, 1, 184614999414571937405905419562270, 249584763877004334779054488506782340719383629107224173, 245395425663663491880846922641400894840783985813370231599231766603156

Offset: 0

Robert A. Russell, Dec 13 2020

Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. For n>120, a(n) = 0.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide generating functions here using bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k.
For the 600 facets of the 600-cell (vertices of the 120-cell), the generating function is 2*bp(20)/15 + bp(30)/5 + 2*bp(40)/15 + bp(50)/6 + 13*bp(60)/150 + bp(100)/180 + bp(104)/18 + 13*bp(120)/150 + bp(150)/120 + bp(200)/180 + bp(208)/18 + bp(300)/7200 + bp(302)/16 + bp(600)/7200.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is 2*bp(24)/15 + bp(36)/5 + 2*bp(48)/15 + bp(60)/6 + 7*bp(72)/150 + bp(76)/25 + 11*bp(120)/180 + 7*bp(144)/150 + bp(152)/25 + bp(180)/120 + 11*bp(240)/180 + bp(360)/7200 + bp(364)/16 + bp(720)/7200.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is 2*bp(40)/15 + bp(60)/5 + 2*bp(80)/15 + bp(100)/6 + 13*bp(120)/150 + bp(200)/180 + bp(202)/18 + 13*bp(240)/150 + bp(300)/120 + bp(400)/180 + bp(404)/18 + bp(600)/7200 + bp(604)/16 + bp(1200)/7200.

Robert A. Russell, Table of n, a(n) for n = 0..120

Cf. A338981 (unoriented), A338982 (chiral), A338983 (achiral), A338964 (up to n colors), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956 (16-cell vertices, 8-cell facets), A338948 (24-cell).

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[2bp[4]/15+bp[6]/5+2bp[8]/15+bp[10]/6+7bp[12]/150+bp[16]/25+bp[20]/180+bp[22]/18+7bp[24]/150+bp[30]/120+bp[32]/25+bp[40]/180+bp[44]/18+bp[60]/7200+bp[62]/16+bp[120]/7200,x]

A338964(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338981(n) + A338982(n) = 2*A338981(n) - A338983(n) = 2*A338982(n) + A338983(n).
G.f.: 2*bp(4)/15 + bp(6)/5 + 2*bp(8)/15 + bp(10)/6 + 7*bp(12)/150 + bp(16)/25 + bp(20)/180 + bp(22)/18 + 7*bp(24)/150 + bp(30)/120 + bp(32)/25 + bp(40)/180 + bp(44)/18 + bp(60)/7200 + bp(62)/16 + bp(120)/7200, where bp(j) = Sum_{k=1..j} k! * S2(j,k) * x^k and S2(j,k) is the Stirling subset number, A008277.

cp's OEIS Frontend

A337956 Number of oriented colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

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