A228695 - OEIS

%I #17 Jan 08 2018 01:47:49

%S 1,1,7,47,521,7233,129443,2811701,73203561,2229207953,78389689559,

%T 3138945552419,141714151130833,7146006410498833,399443567886826899,

%U 24581290495461129817,1655664011866577666737,121413069330848040859809,9648772995329567310573319

%N Number of labeled graphs on 2n nodes with degree set {1,2,3}, with multiple edges and loops allowed.

%H Vaclav Kotesovec, <a href="/A228695/b228695.txt">Table of n, a(n) for n = 0..320</a>

%H I. P. Goulden and D. M. Jackson, <a href="http://dx.doi.org/10.1137/0607007">Labelled graphs with small vertex degrees and P-recursiveness</a>, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093).

%F See Goulden-Jackson for the e.g.f.

%F Recurrence (for n>9): 12*(3*n^4 - 19*n^3 + 19*n^2 + 24*n - 31)*a(n) = 6*(9*n^5 - 57*n^4 + 35*n^3 + 160*n^2 - 151*n - 4)*a(n-1) + 9*(n-1)*(3*n^6 - 25*n^5 + 61*n^4 - 16*n^3 - 135*n^2 + 104*n - 4)*a(n-2) + 3*(n-2)*(n-1)*(21*n^5 - 106*n^4 - 62*n^3 + 603*n^2 - 448*n - 6)*a(n-3) + 3*(n-3)*(n-2)*(n-1)*(21*n^5 - 106*n^4 + 15*n^3 + 208*n^2 - 209*n - 46)*a(n-4) + (n-4)*(n-3)*(n-2)*(n-1)*(51*n^4 - 77*n^3 - 526*n^2 + 477*n - 110)*a(n-5) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n^5 - 42*n^4 - 29*n^3 + 159*n^2 - 120*n + 30)*a(n-6) - (n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(6*n^4 - 14*n^3 - 11*n^2 + 22*n + 10)*a(n-7) + (n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(3*n^4 - 7*n^3 - 20*n^2 + 17*n - 4)*a(n-8). - _Vaclav Kotesovec_, Sep 15 2014

%t max=20; f[x_]:=Sum[a[n]*(x^(n)/n!),{n,0,max}]; a[0]=1; a[1]=1; coef = CoefficientList[9*x^3*(x+2)*(x^3 - 2*x^2 + x - 1)*f''[x] - 3*(x^10 - 10*x^8 - 6*x^7 + 22*x^6 + 8*x^5 + 20*x^4 + 26*x^3 + 16*x - 8)*f'[x] + (x^11 - 2*x^10 - 14*x^9 + 24*x^8 + 74*x^7 - 61*x^6 - 99*x^5 - 55*x^4 - 180*x^3 - 48*x^2 - 96*x - 24)*f[x],x]; Table[a[n],{n,0,max}]/.Solve[Thread[coef[[2;;max]]==0]][[1]] (* _Vaclav Kotesovec_, Sep 15 2014 *)

%Y Cf. A005814, A188404, A228694.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Sep 02 2013

%E More terms from _Vaclav Kotesovec_, Sep 15 2014