This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A063843 #43 Dec 28 2020 17:45:10
%S A063843 0,1,34,792,10688,90005,533358,2437848,9156288,29522961,84293770,
%T A063843 217993600,519341472,1154658869,2420188694,4821091920,9187076352,
%U A063843 16837177281,29809183410,51172613512,85448030080,139159855989,221554769150,345523218536,528767663040
%N A063843 Number of n-multigraphs on 5 nodes.
%C A063843 Equivalently, number of ways to color edges of complete graph on 5 nodes with n colors, under action of symmetric group S_5, of order 120, with cycle index on edges given by (1/120)*(24*x5^2 + 30*x2*x4^2 + 20*x3^3*x1 + 20*x3*x6*x1 + 15*x1^2*x2^4 + 10*x1^4*x2^3 + x1^10). Setting all x_i = n gives the sequence.
%C A063843 Number of vertex colorings of the Petersen graph. _Marko Riedel_, Mar 24 2016
%C A063843 Number of unoriented colorings of the 10 triangular edges or triangular faces of a pentachoron, Schläfli symbol {3,3,3}, using n or fewer colors. Also called a 5-cell or 4-simplex. - _Robert A. Russell_, Oct 17 2020
%H A063843 T. D. Noe, <a href="/A063843/b063843.txt">Table of n, a(n) for n = 0..1000</a>
%H A063843 Vladeta Jovovic, <a href="/A063843/a063843.rtf">Formulae for the number T(n,k) of n-multigraphs on k nodes</a>
%H A063843 Marko Riedel, <a href="http://math.stackexchange.com/questions/1711016/">Vertex colorings of the Petersen graph</a>, Math StackExchange
%H A063843 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A063843 a(n) = (1/120)*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10).
%F A063843 a(n+1) = (1/5!)*(n^10 + 10*n^9 + 45*n^8 + 130*n^7 + 295*n^6 + 552*n^5 + 805*n^4 + 900*n^3 + 774*n^2 + 448*n + 120).
%F A063843 G.f. = (1 + 23*x + 473*x^2 + 3681*x^3 + 10717*x^4 + 11221*x^5 + 3779*x^6 + 339*x^7 + 6*x^8)/(1-x)^11. - _M. F. Hasler_, Jan 19 2012
%F A063843 a(0)=0, a(1)=1, a(2)=34, a(3)=792, a(4)=10688, a(5)=90005, a(6)=533358, a(7)=2437848, a(8)=9156288, a(9)=29522961, a(10)=84293770, 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). - _Harvey P. Dale_, Oct 20 2012
%F A063843 From _Robert A. Russell_, Oct 17 2020: (Start)
%F A063843 a(n) = A331350(n) - A331352(n) = (A331350(n) + A331353(n)) / 2 = A331352(n) + A331353(n).
%F A063843 a(n) = 1*C(n,1) + 32*C(n,2) + 693*C(n,3) + 7720*C(n,4) + 44150*C(n,5) + 138312*C(n,6) + 247380*C(n,7) + 252000*C(n,8) + 136080*C(n,9) + 30240*C(n,10), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors. (End)
%p A063843 f:=n-> 1/120*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10);
%t A063843 Table[(24n^2+50n^3+20n^4+15n^6+10n^7+n^10)/120,{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,1,34,792,10688,90005,533358,2437848,9156288,29522961,84293770},30] (* _Harvey P. Dale_, Oct 20 2012 *)
%o A063843 (PARI) a(n)=n^2*(n^8+10*n^5+15*n^4+20*n^2+50*n+24)/120 \\ _Charles R Greathouse IV_, Jan 20 2012
%Y A063843 Cf. A063842. A row of A063841.
%Y A063843 Cf. A331350 (oriented), A331352 (chiral), A331353 (achiral), A000389(n+4) (vertices and facets)
%Y A063843 Other polychora: A331359 (8-cell), A331355 (16-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).
%Y A063843 Row 4 of A327084 (simplex edges and ridges) and A337884 (simplex faces and peaks).
%K A063843 nonn,easy
%O A063843 0,3
%A A063843 _N. J. A. Sloane_, Aug 25 2001
%E A063843 More terms from _Vladeta Jovovic_, Sep 02 2001