A228694 Number of labeled graphs on 2n nodes with degree set {1,3}, with multiple edges and loops allowed.
1, 5, 186, 22960, 6831650, 4071581010, 4297593045900, 7359945086654160, 19160998099781838300, 72124861521922576867500, 377272837054974521764903800, 2655805439512625993259947280000, 24502785480337947107875310460499800, 289788471352423824164622588783247815000
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..160
- 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).
Programs
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Mathematica
max=30; f[x_]:=Sum[a[n]*(x^n/n!),{n,0,max}]; a[0]=1; a[1]=5; coef = CoefficientList[9*x^3*(x^4 - 4*x^2 - 2)*f''[x] - 3*(x^10 - 14*x^8 + 41*x^6 + 36*x^4 + 2*x^2 - 8)*f'[x] + x*(x^10 - 18*x^8 + 120*x^6 - 272*x^4 - 324*x^2 - 120)*f[x],x]; Table[a[n],{n,0,max,2}]/.Solve[Thread[coef[[2;;max]]==0]][[1]] (* Vaclav Kotesovec, Sep 15 2014 *)
Formula
See Goulden-Jackson for the e.g.f.
Recurrence (for n>5): 3*a(n) = 3*(2*n - 1)*(3*n^2 - 3*n + 1)*a(n-1) + 3*(n-1)*(2*n - 3)*(2*n - 1)*(10*n + 7)*a(n-2) - 2*(n-2)*(n-1)*(2*n - 5)*(2*n - 3)*(2*n - 1)*(3*n + 4)*a(n-3) + 2*(n-3)*(n-2)*(n+1)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)*a(n-4). - Vaclav Kotesovec, Sep 15 2014
a(n) ~ sqrt(2) * 6^n * n^(3*n) / exp(3*n-4). - Vaclav Kotesovec, Sep 15 2014
Extensions
More terms from Vaclav Kotesovec, Sep 15 2014