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A338981
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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.
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6
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0, 1, 92307499707443390526727850063502, 124792381938502167392061689732085833655832902312754962, 122697712831831745940423467267565845711242845618544066030140191642464
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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A338965(n) = Sum_{j=1..Min(n,120)} a(n) * binomial(n,j).
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.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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fini,nonn,easy
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AUTHOR
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STATUS
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approved
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