A063842 Number of colorings of K_4 using at most n colors.

1, 11, 66, 276, 900, 2451, 5831, 12496, 24651, 45475, 79376, 132276, 211926, 328251, 493725, 723776, 1037221, 1456731, 2009326, 2726900, 3646776, 4812291, 6273411, 8087376, 10319375, 13043251, 16342236, 20309716, 25050026, 30679275, 37326201, 45133056

Offset: 0

Author

N. J. A. Sloane, Aug 25 2001

Keywords

• nonn
• easy

Comments

a(n-1) is the number of unoriented ways to color the edges of a regular tetrahedron with up to n colors.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
• Marko R. Riedel, Counting multigraphs up to isomorphism
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

A row of A063841. Cf. A063843.

A327084(3,n) = a(n-1) (unoriented simplex edge colorings)

Programs

Magma

[1/24*(n^6+6*n^5+24*n^4+56*n^3+83*n^2+70*n+24): n in [0..35]]; // Vincenzo Librandi, Jul 21 2013

Mathematica

Needs["Combinatorica`"]
Table[Total[Table[CycleIndex[KSubsetGroup[GraphData[{4,k},"Automorphisms"],GraphData[{4,k},"EdgeIndices"]],s],{k,1,11}]]/.Table[s[i] -> n,{i,1,4}],{n,0,30}]  (* Geoffrey Critzer, Oct 22 2012 *)
CoefficientList[Series[(1 + 3 x + 7 x^2 + 3 x^3 + x^4) (1 + x) / (1 - x)^7, {x, 0, 35}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,11,66,276,900,2451,5831},40] (* Harvey P. Dale, Sep 10 2023 *)

Formula

a(n) = (1/4!)*(n^6 + 6*n^5 + 24*n^4 + 56*n^3 + 83*n^2 + 70*n + 24).

G.f.: (1 + 3*x + 7*x^2 + 3*x^3 + x^4)*(1+x)/(1-x)^7. - M. F. Hasler, Jan 19 2012

Extensions

More terms from Vladeta Jovovic, Sep 02 2001