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

%I A063842 #39 Sep 10 2023 12:04:08
%S A063842 1,11,66,276,900,2451,5831,12496,24651,45475,79376,132276,211926,
%T A063842 328251,493725,723776,1037221,1456731,2009326,2726900,3646776,4812291,
%U A063842 6273411,8087376,10319375,13043251,16342236,20309716,25050026,30679275,37326201,45133056
%N A063842 Number of colorings of K_4 using at most n colors.
%C A063842 a(n-1) is the number of unoriented ways to color the edges of a regular tetrahedron with up to n colors.
%H A063842 Vincenzo Librandi, <a href="/A063842/b063842.txt">Table of n, a(n) for n = 0..1000</a>
%H A063842 Vladeta Jovovic, <a href="/A063843/a063843.rtf">Formulae for the number T(n,k) of n-multigraphs on k nodes</a>
%H A063842 Marko R. Riedel, <a href="http://math.stackexchange.com/questions/2088991/">Counting multigraphs up to isomorphism</a>
%H A063842 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A063842 a(n) = (1/4!)*(n^6 + 6*n^5 + 24*n^4 + 56*n^3 + 83*n^2 + 70*n + 24).
%F A063842 G.f.: (1 + 3*x + 7*x^2 + 3*x^3 + x^4)*(1+x)/(1-x)^7. - _M. F. Hasler_, Jan 19 2012
%t A063842 Needs["Combinatorica`"]
%t A063842 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 *)
%t A063842 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 *)
%t A063842 LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,11,66,276,900,2451,5831},40] (* _Harvey P. Dale_, Sep 10 2023 *)
%o A063842 (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
%Y A063842 A row of A063841. Cf. A063843.
%Y A063842 A327084(3,n) = a(n-1) (unoriented simplex edge colorings)
%K A063842 nonn,easy
%O A063842 0,2
%A A063842 _N. J. A. Sloane_, Aug 25 2001
%E A063842 More terms from _Vladeta Jovovic_, Sep 02 2001