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

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

Author

N. J. A. Sloane, Dec 22 2017

Keywords

• nonn
• easy

Links

• 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).

Crossrefs

Cf. A159233, A296912.

Programs

Mathematica

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 *)

PARI

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

Formula

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)