A338965 -id:A338965 - cp's OEIS frontend

A337957 Number of unoriented colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

1, 15, 126, 715, 3060, 10626, 31465, 82251, 194580, 424270, 864501, 1663740, 3049501, 5359095, 9078630, 14891626, 23738715, 36890001, 56031760, 83369265, 121747626, 174792640, 247073751, 344291325, 473490550, 643304376

Offset: 1

Robert A. Russell, Oct 03 2020

Each chiral pair is counted as one when enumerating unoriented 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. A337956 (oriented), A234249(n+1) (chiral), A337958 (achiral).
Other elements: A331355 (hyperoctahedron edges, tesseract faces), A331359 (hyperoctahedron faces, tesseract edges), A128767 (hyperoctahedron facets, tesseract vertices).
Other polychora: A000389(n+4) (5-cell), A338949 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A325005 (orthotope facets, orthoplex vertices).

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

a(n) = binomial(binomial(n+1,2)+3,4).
a(n) = n * (n+1) * (n^2 + n + 2) * (n^2 + n + 4) * (n^2 + n + 6) / 384.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 297*C(n,4) + 600*C(n,5) + 690*C(n,6) + 420*C(n,7) + 105*C(n,8), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A337956(n) - A234249(n+1) = (A337956(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 + 37*x^3 + 27*x^4 + 6*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)

A338981 Number of unoriented 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, 92307499707443390526727850063502, 124792381938502167392061689732085833655832902312754962, 122697712831831745940423467267565845711242845618544066030140191642464

Offset: 0

Robert A. Russell, Dec 13 2020

Each chiral pair is counted as one when enumerating unoriented 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 bp(20)/15 + bp(30)/10 + bp(40)/15 + bp(50)/12 + 43*bp(60)/300 + bp(66)/10 + bp(100)/360 + bp(104)/9 + bp(114)/12 + 13*bp(120)/300 + bp(150)/240 + bp(152)/8 + bp(200)/360 + bp(208)/36 + 61*bp(300)/14400 + bp(302)/32 + bp(330)/240 + bp(600)/14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the generating function is bp(24)/15 + bp(36)/10 + bp(48)/15 + bp(60)/12 + 7*bp(72)/300 + 3*bp(76)/25 + bp(84)/10 + 41*bp(120)/360 + bp(132)/12 + 7*bp(144)/300 + bp(152)/50 + bp(180)/240 + bp(182)/8 + 11*bp(240)/360 + 61*bp(360)/14400 + bp(364)/32 + bp(396)/240 + bp(720)/14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the generating function is bp(40)/15 + bp(60)/10 + bp(80)/15 + bp(100)/12 + 43*bp(120)/300 + bp(128)/10 + bp(200)/360 + bp(202)/9 + bp(216)/12 + 13*bp(240)/300 + bp(300)/240 + bp(302)/8 + bp(400)/360 + bp(404)/36 + 61*bp(600)/14400 + bp(604)/32 + bp(640)/240 + bp(1200)/14400.

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

Cf. A338980 (oriented), A338982 (chiral), A338983 (achiral), A338965 (up to n colors), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338949 (24-cell).

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, j}] (*binomial series*)
CoefficientList[bp[4]/15+bp[6]/10+bp[8]/15+bp[10]/12+7bp[12]/300+bp[16]/50+bp[17]/10+bp[19]/10+bp[20]/360+bp[22]/36+bp[23]/12+7bp[24]/300+bp[27]/12+bp[30]/240+bp[31]/8+bp[32]/50+bp[40]/360+bp[44]/36+bp[60]/14400+bp[61]/240+bp[62]/32+bp[75]/240+bp[120]/14400,x]

A338965(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
a(n) = A338980(n) - A338982(n) = (A338980(n) + A338983(n)) / 2 = A338982(n) + A338983(n).
G.f.: bp(4)/15 + bp(6)/10 + bp(8)/15 + bp(10)/12 + 7bp(12)/300 + bp(16)/50 + bp(17)/10 + bp(19)/10 + bp(20)/360 + bp(22)/36 + bp(23)/12 + 7bp(24)/300 + bp(27)/12 + bp(30)/240 + bp(31)/8 + bp(32)/50 + bp(40)/360 + bp(44)/36 + bp(60)/14400 + bp(61)/240 + bp(62)/32 + bp(75)/240 + bp(120)/14400, 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

A337957 Number of unoriented colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

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