A228696 Number of labeled graphs on n nodes with degree set {2,4}, with multiple edges and loops allowed.

1, 2, 9, 65, 751, 13044, 320803, 10609256, 453774440, 24375801464, 1607240682376, 127684970262822, 12034618723574314, 1328262275098167080, 169754658940294717086, 24877923644434862091314, 4145248921431765036724198, 779383143209936088442516156, 164246020015613168238167009350

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

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>11): 96*(2*n^5 - 8*n^4 - 3*n^3 + 22*n^2 + 69*n - 88)*a(n) = 32*(4*n^7 - 26*n^6 + 34*n^5 + 29*n^4 + 121*n^3 - 554*n^2 + 350*n + 6)*a(n-1) + 32*(n-1)*(4*n^7 + 10*n^6 - 90*n^5 - 130*n^4 + 705*n^3 + 587*n^2 - 1286*n + 26)*a(n-2) - 16*(n-2)*(n-1)*(6*n^7 + 2*n^6 - 151*n^5 - 13*n^4 + 687*n^3 + 1225*n^2 - 2200*n - 64)*a(n-3) - 8*(n-3)*(n-2)*(n-1)*(8*n^7 - 34*n^6 + 96*n^5 + 43*n^4 - 480*n^3 + 249*n^2 + 1524*n - 208)*a(n-4) + 8*(n-4)*(n-3)*(n-2)*(n-1)*(4*n^7 - 58*n^6 + 98*n^5 + 417*n^4 - 621*n^3 - 2092*n^2 + 2042*n + 106)*a(n-5) + 8*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(12*n^6 - 52*n^5 - 167*n^4 + 453*n^3 + 850*n^2 - 1894*n + 58)*a(n-6) + 2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(8*n^6 + 6*n^5 - 12*n^4 - 11*n^3 - 306*n^2 + 1325*n - 160)*a(n-7) - 2*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(4*n^6 - 16*n^5 - 42*n^4 + 112*n^3 + 340*n^2 - 633*n + 38)*a(n-8) - 2*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(4*n^5 + 2*n^4 - 28*n^3 - 14*n^2 + 54*n + 5)*a(n-9) + (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n^5 + 2*n^4 - 15*n^3 - 15*n^2 + 82*n - 6)*a(n-10). - Vaclav Kotesovec, Sep 15 2014

Mathematica

max = 20; f[x_] := Sum[a[n]*(x^n/n!), {n, 0, max}]; a[0] = 1; a[1] = 2; coef = CoefficientList[-16*(x - 2)^2* x^2*(x + 1)^2*(x^5 - 2*x^4 + 2*x^3 - 2*x^2 + 12*x + 4)*f''[x] + 4*(x^13 - 4*x^12 - 6*x^11 + 36*x^10 - 6*x^9 + 24*x^8 - 352*x^7 + 380*x^6 + 152*x^5 + 2104*x^4 - 1472*x^3 - 688*x^2 + 256*x + 96)* f'[x] + (-x^14 + 6*x^13 + 2*x^12 - 76*x^11 + 112*x^10 + 96*x^9 + 356*x^8 - 1320*x^7 - 568*x^6 + 768*x^5 + 9248*x^4 + 12224*x^3 - 2496*x^2 - 3968*x - 768)*f[x], x]; Table[a[n], {n, 0, max}] /. Solve[Thread[coef[[2 ;; max]] == 0]][[1]] (* Vaclav Kotesovec, Sep 15 2014 *)

Crossrefs

Cf. A228697.

Sequence in context: A334315 A334263 A127056 * A042255 A152213 A259607

Adjacent sequences: A228693 A228694 A228695 * A228697 A228698 A228699

Keyword

nonn

Author

N. J. A. Sloane, Sep 02 2013

Extensions

More terms from Vaclav Kotesovec, Sep 15 2014

Status

approved