A228695 Number of labeled graphs on 2n nodes with degree set {1,2,3}, with multiple edges and loops allowed.

1, 1, 7, 47, 521, 7233, 129443, 2811701, 73203561, 2229207953, 78389689559, 3138945552419, 141714151130833, 7146006410498833, 399443567886826899, 24581290495461129817, 1655664011866577666737, 121413069330848040859809, 9648772995329567310573319

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..320

I. P. Goulden and D. M. Jackson, Labelled graphs with small vertex degrees and P-recursiveness, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093).

Formula

See Goulden-Jackson for the e.g.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

Mathematica

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 *)

Crossrefs

Cf. A005814, A188404, A228694.

Sequence in context: A178002 A288722 A006873 * A368295 A268063 A015097

Adjacent sequences: A228692 A228693 A228694 * A228696 A228697 A228698

Keyword

nonn

Author

N. J. A. Sloane, Sep 02 2013

Extensions

More terms from Vaclav Kotesovec, Sep 15 2014

Status

approved