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A282670 Number of inequivalent ways to color the edges of a dodecahedron using at most n colors.

%I A282670 #17 Mar 09 2024 11:34:20
%S A282670 0,1,17912448,3431529649899,19215359484207104,15522042948408209375,
%T A282670 3684565329384186949248,375655671519845961645597,
%U A282670 20632333988160040350515200,706519304587399981447927557,16666666666669166670000400000,290823371148118276083759139095
%N A282670 Number of inequivalent ways to color the edges of a dodecahedron using at most n colors.
%C A282670 Cycle index of symmetry group A5 acting on the 30 edges of the dodecahedron is (24s(5)^6 + 20s(3)^10 + 15s(2)^14*s(1)^2 + s(1)^30)/60.
%C A282670 Also the number of inequivalent ways to color the edges of the icosahedron using at most n colors.
%C A282670 From _Robert A. Russell_, Oct 03 2020: (Start)
%C A282670 Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual. There are 60 elements in the rotation group of the regular dodecahedron/icosahedron. They divide into five conjugacy classes. The first formula is obtained by averaging the edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A282670   Conjugacy Class    Count    Even Cycle Indices
%C A282670   Identity              1     x_1^30
%C A282670   Edge rotation        15     x_1^2x_2^14
%C A282670   Vertex rotation      20     x_3^10
%C A282670   Small face rotation  12     x_5^6
%C A282670 Large face rotation  12     x_5^6  (End)
%H A282670 <a href="/index/Rec#order_31">Index entries for linear recurrences with constant coefficients</a>, signature (31, -465, 4495, -31465, 169911, -736281, 2629575, -7888725, 20160075, -44352165, 84672315, -141120525, 206253075, -265182525, 300540195, -300540195, 265182525, -206253075, 141120525, -84672315, 44352165, -20160075, 7888725, -2629575, 736281, -169911, 31465, -4495, 465, -31, 1).
%F A282670 a(n) = n^6 (n^24 + 15 n^10 + 20 n^4 + 24)/60.
%F A282670 G.f.: x*(1 + x)*(1 + 17912416*x + 3430956452060*x^2 + 19105559437892000*x^3 + 14908856825730677891*x^4 + 3197392859155796794496*x^5 + 265368238349945588707496*x^6 + 10365795256050146806088576*x^7 + 215154060506484358838662001*x^8 + 2568188846096433625477331936*x^9 + 18582986600475456162494990756*x^10 + 84400699070086923625163495456*x^11 + 245956255494355672481225103371*x^12 + 465612713610802763378946154496*x^13 + 575747234318647571242943474096*x^14 + 465612713610802763378946154496*x^15 + 245956255494355672481225103371*x^16 + 84400699070086923625163495456*x^17 + 18582986600475456162494990756*x^18 + 2568188846096433625477331936*x^19 + 215154060506484358838662001*x^20 + 10365795256050146806088576*x^21 + 265368238349945588707496*x^22 + 3197392859155796794496*x^23 + 14908856825730677891*x^24 + 19105559437892000*x^25 + 3430956452060*x^26 + 17912416*x^27 + x^28) / (1 - x)^31. - _Colin Barker_, Mar 30 2019
%F A282670 From _Robert A. Russell_, Oct 03 2020: (Start)
%F A282670 a(n) = 1*C(n,1) + 17912446*C(n,2) + 3431475912558*C(n,3) + 19201633473082192*C(n,4) + 15426000466104548370*C(n,5) + 3591721233455676488292*C(n,6) + 350189004698594439734160*C(n,7) + 17729388555701917767855840*C(n,8) + 534044352737570253478824960*C(n,9) + 10485619820879148545218980480*C(n,10) + 143066535726280748444739676800*C(n,11) + 1420876074163106703694904352000*C(n,12) + 10631861498419617103267350931200*C(n,13) + 61515486939441778743810979468800*C(n,14) + 280711222366395106969585943040000*C(n,15) + 1025499893865270227589218761728000*C(n,16) + 3032858772294885663526454593536000*C(n,17) + 7319173455487770465200322686976000*C(n,18) + 14487618384525410959295952691200000*C(n,19) + 23580333216029318427870396825600000*C(n,20) + 31555723729541430372276884520960000*C(n,21) + 34619561317726617824610327429120000*C(n,22) + 30946535969611314628728933580800000*C(n,23) + 22311118596400512968549479219200000*C(n,24) + 12771433990957347267674112000000000*C(n,25) + 5668281691036644651075462758400000*C(n,26) + 1879979643918904128084836352000000*C(n,27) + 438404032189593555246120960000000*C(n,28) + 64102774454612839170441216000000*C(n,29) + 4420880996869850977271808000000*C(n,30), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
%F A282670 a(n) = A337963(n) + A337964(n) = 2*A337963(n) - A337953(n) = 2*A337964(n) + A337953(n). (End)
%e A282670 There are a(2) = 17912448 inequivalent ways to color the edges of the dodecahedron using at most two colors.
%t A282670 Table[(24n^6+20n^10+15n^16+n^30)/60, {n, 0, 16}]
%Y A282670 Other elements: A054472 (dodecahedron vertices, icosahedron faces), A000545 (dodecahedron faces, icosahedron vertices).
%Y A282670 Other polyhedra: A046023 (tetrahedron), A060530 (cube/octahedron).
%Y A282670 Cf. A337963 (unoriented), A337964 (chiral), A337953 (achiral).
%K A282670 easy,nonn
%O A282670 0,3
%A A282670 _David Nacin_, Feb 20 2017