A159233 -id:A159233 - cp's OEIS frontend

A270842 Number of nonisomorphic edge colorings of the Petersen graph with at most n colors.

1, 396, 123786, 9002912, 254721400, 3920311044, 39571426713, 293231076608, 1715840171595, 8333541708700, 34810892718492, 128392921513440, 426551317876970, 1296405100924948, 3649123762524675, 9607693522053120, 23853550135649477, 56222046462953772

Offset: 1

Marko Riedel, Mar 24 2016

These are not colorings in the strict sense, since there is no requirement that adjacent edges have different colors. - N. J. A. Sloane, Mar 28 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Math StackExchange, Edge colorings of the Petersen graph

Cf. A270843, A063843. See A159233 for edge colorings where adjacent edges must have different colors.

a(n) = n^15/120 + 5*n^9/24 + 5*n^5/12 + 11*n^3/30; \\ Altug Alkan, Mar 25 2016

Cycle index of the automorphisms acting on the edges is (1/120)*S[1]^15+(5/24)*S[2]^6*S[1]^3+(1/4)*S[4]^3*S[2]*S[1]+(1/6)*S[3]^5+(1/6)*S[3]*S[6]^2+(1/5)*S[5]^3.
a(n) = n^15/120 + 5*n^9/24 + 5*n^5/12 + 11*n^3/30.
G.f.: x*(1 + 380*x + 117570*x^2 + 7069296*x^3 + 125309188*x^4 + 856514276*x^5 + 2594956089*x^6 + 3729352800*x^7 + 2594956089*x^8 + 856514276*x^9 + 125309188*x^10 + 7069296*x^11 + 117570*x^12 + 380*x^13 + x^14) / (1 - x)^16. - Colin Barker, Dec 24 2017

A296913 Number of ways to properly color the Petersen graph using n colors.

Original entry on oeis.org

0, 0, 120, 12960, 332880, 3868080, 27767880, 144278400, 594347040, 2055598560, 6202551960, 16774966560, 41473626480, 95135323920, 204803912040, 417515696640, 811858751040, 1514650599360, 2724410748600, 4743687388320, 8022734847120, 13217533726320, 21265702652040, 33484472926080, 51695588642400, 78382758698400, 116888127197400

Offset: 1

N. J. A. Sloane, Dec 22 2017

Michael De Vlieger, Table of n, a(n) for n = 1..10000
M. Baker, Hodge Theory in Combinatorics, Bull. Amer. Math. Soc., 55 (No. 1, 2018), 57-80. See p. 60.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A159233, A296912.

Array[#^10 - 15 #^9 + 105 #^8 - 455 #^7 + 1353 #^6 - 2861 #^5 + 4275 #^4 - 4305 #^3 + 2606 #^2 - 704 # &, 27] (* Michael De Vlieger, Dec 23 2017 *)
Rest@ CoefficientList[ Series[-120 x^3 (139x^7 +1693x^6 +7269x^5 +11905x^4 +7495x^3 +1641x^2 +97x +1)/(x -1)^11, {x, 0, 23}], x] (* or *)
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 0, 120, 12960, 332880, 3868080, 27767880, 144278400, 594347040, 2055598560, 6202551960}, 23] (* Robert G. Wilson v, Dec 24 2017 *)

concat(vector(2), Vec(120*x^3*(1 + 97*x + 1641*x^2 + 7495*x^3 + 11905*x^4 + 7269*x^5 + 1693*x^6 + 139*x^7) / (1 - x)^11 + O(x^40))) \\ Colin Barker, Dec 24 2017

a(n) = n^10 - 15*n^9 + 105*n^8 - 455*n^7 + 1353*n^6 - 2861*n^5 + 4275*n^4 - 4305*n^3 + 2606*n^2 - 704*n.
From Colin Barker, Dec 24 2017: (Start)
G.f.: 120*x^3*(1 + 97*x + 1641*x^2 + 7495*x^3 + 11905*x^4 + 7269*x^5 + 1693*x^6 + 139*x^7) / (1 - x)^11.
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>11.
(End)

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A270842 Number of nonisomorphic edge colorings of the Petersen graph with at most n colors.

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