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%I #9 Mar 09 2024 11:36:11
%S 1,8972888,1715781087090,9607681898535232,7761021569825850025,
%T 1842282666811844114760,187827835789041358086652,
%U 10316166994361788355074560,353259652295786354195866209
%N Number of unoriented colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.
%C Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.
%H <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 a(n) = (n^30 + 15*n^17 + 15*n^16 + n^15 + 20*n^10 + 24*n^6 + 20*n^5 + 24*n^3) / 120.
%F a(n) = 1*C(n,1) + 8972886*C(n,2) + 1715754168429*C(n,3) + 9600818828024196*C(n,4) + 7713000318054315890*C(n,5) + 1795860618305879894604*C(n,6) + 175094502365510493018246*C(n,7) + 8864694277953928285823280*C(n,8) + 267022176369217557115630320*C(n,9) + 5242809910440825835898466240*C(n,10) + 71533267863142929693959229120*C(n,11) + 710438037081557065871500310400*C(n,12) + 5315930749209812373842350550400*C(n,13) + 30757743469720892095213642099200*C(n,14) + 140355611183197554763055563526400*C(n,15) + 512749946932635114150296808960000*C(n,16) + 1516429386147442831807688225280000*C(n,17) + 3659586727743885232600161343488000*C(n,18) + 7243809192262705479647976345600000*C(n,19) + 11790166608014659213935198412800000*C(n,20) + 15777861864770715186138442260480000*C(n,21) + 17309780658863308912305163714560000*C(n,22) + 15473267984805657314364466790400000*C(n,23) + 11155559298200256484274739609600000*C(n,24) + 6385716995478673633837056000000000*C(n,25) + 2834140845518322325537731379200000*C(n,26) + 939989821959452064042418176000000*C(n,27) + 219202016094796777623060480000000*C(n,28) + 32051387227306419585220608000000*C(n,29) + 2210440498434925488635904000000*C(n,30), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
%F a(n) = A282670(n) - A337964(n) = (A282670(n) + A337953(n)) / 2 = A337964(n) + A337953(n).
%t Table[(n^30+15n^17+15n^16+n^15+20n^10+24n^6+20n^5+24 n^3)/120,{n,30}]
%Y Cf. A282670 (oriented), A337964 (chiral), A337953 (achiral).
%Y Other elements: A252704 (dodecahedron vertices, icosahedron faces), A252705 (dodecahedron faces, icosahedron vertices).
%Y Other polyhedra: A063842(n-1) (tetrahedron), A199406 (cube/octahedron).
%K nonn,easy
%O 1,2
%A _Robert A. Russell_, Oct 03 2020
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