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%I #19 Mar 16 2020 20:14:14
%S 1,2,10,31,90,222,520,1090,2180,4090,7356,12660,21105,34020,53460,
%T 81891,122826,180510,260746,370370,518518,715870,976170,1315470,
%U 1753975,2314936,3027224,3923845,5044920,6436200,8152542,10255896
%N Number of directed multigraphs with loops on 3 nodes with n arcs.
%H Andrew Howroyd, <a href="/A050927/b050927.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).
%F G.f.: (x^10+3*x^8+10*x^7+16*x^6+12*x^5+16*x^4+10*x^3+3*x^2+1) / ((1-x^3)^3*(1-x^2)^4*(1-x)^2).
%t <<Combinatorica`
%t nn=30;n=3;CoefficientList[Series[CycleIndex[Join[PairGroup[SymmetricGroup[n],Ordered],Permutations[Range[n*(n - 1) + 1, n*(n - 1) + n]], 2], s] /.Table[s[i] -> 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* _Geoffrey Critzer_, Aug 07 2015 *)
%t CoefficientList[Series[(x^10 + 3 x^8 + 10 x^7 + 16 x^6 + 12 x^5 + 16 x^4 + 10 x^3 + 3 x^2 + 1)/((1 - x^3)^3 (1 - x^2)^4 (1 - x)^2), {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 08 2015 *)
%o (PARI) Vec((1 + 3*x^2 + 10*x^3 + 16*x^4 + 12*x^5 + 16*x^6 + 10*x^7 + 3*x^8 + x^10)/((1 - x)^2*(1 - x^2)^4*(1 - x^3)^3) + O(x^40)) \\ _Andrew Howroyd_, Mar 16 2020
%Y Column k=3 of A138107.
%Y Cf. A005993.
%K easy,nonn
%O 0,2
%A _Vladeta Jovovic_, Dec 30 1999
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