A228697 Number of labeled graphs on n nodes with degree set {2,4}.

1, 0, 0, 1, 3, 38, 730, 20670, 781578, 37885204, 2289786624, 168879532980, 14930452838620, 1558773675667164, 189751596502711412, 26640166205940265342, 4272887377701537747810, 776487144927243427031040, 158705790134274513564693472, 36246543562488282624570636306

Offset

0,5

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*(72*n^5 - 864*n^4 + 3804*n^3 - 7474*n^2 + 6468*n - 1807)*a(n) = 32*(n-1)*(144*n^6 - 1512*n^5 + 5232*n^4 - 5804*n^3 - 1566*n^2 + 3538*n + 2577)*a(n-1) - 32*(n-2)*(n-1)*(144*n^6 - 1944*n^5 + 10056*n^4 - 24152*n^3 + 23988*n^2 - 4910*n - 1403)*a(n-2) - 16*(n-2)*(n-1)*(216*n^7 - 3096*n^6 + 16524*n^5 - 39258*n^4 + 33862*n^3 + 9987*n^2 - 17493*n - 4173)*a(n-3) + 8*(n-3)*(n-2)*(n-1)*(288*n^7 - 5112*n^6 + 35736*n^5 - 121996*n^4 + 198314*n^3 - 103016*n^2 - 55443*n + 36000)*a(n-4) + 8*(n-4)*(n-3)*(n-2)*(n-1)*(144*n^7 - 2232*n^6 + 13008*n^5 - 33968*n^4 + 32794*n^3 + 8104*n^2 - 19105*n + 6412)*a(n-5) + 8*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(144*n^6 - 1872*n^5 + 8472*n^4 - 14732*n^3 + 5270*n^2 + 6047*n - 1153)*a(n-6) - 2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(288*n^6 - 4248*n^5 + 21840*n^4 - 47548*n^3 + 33662*n^2 + 13820*n - 10493)*a(n-7) - 4*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(72*n^6 - 936*n^5 + 4092*n^4 - 6628*n^3 + 2284*n^2 + 1144*n - 737)*a(n-8) - 2*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(144*n^5 - 1080*n^4 + 2712*n^3 - 1946*n^2 - 810*n + 719)*a(n-9) - (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(72*n^5 - 504*n^4 + 1068*n^3 - 526*n^2 - 164*n + 199)*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] = 0; coef = CoefficientList[-16*(x - 1)^2* x^2*(x + 2)^2*(x^2 + 2*x - 2)*(x^3 + 2)*f''[x] + 4*(x^13 + 4*x^12 - 2*x^11 - 20*x^10 + 2*x^9 + 40*x^8 - 104*x^7 - 204*x^6 + 200*x^5 + 328*x^4 - 288*x^3 - 208*x^2 + 320*x - 96)* f'[x] + x^2*(x^12 + 6*x^11 + 14*x^10 + 12*x^9 - 16*x^8 + 24*x^7 + 116*x^6 - 184*x^5 + -456*x^4 + 480*x^3 + 512*x^2 - 704*x + 192)* f[x], x]; Table[a[n], {n, 0, max}] /. Solve[Thread[coef[[2 ;; max]] == 0]][[1]] (* Vaclav Kotesovec, Sep 15 2014 *)

Crossrefs

Cf. A228696.

Sequence in context: A265914 A005780 A033678 * A072331 A109518 A300631

Adjacent sequences: A228694 A228695 A228696 * A228698 A228699 A228700

Keyword

nonn

Author

N. J. A. Sloane, Sep 02 2013

Extensions

More terms from Vaclav Kotesovec, Sep 15 2014

Status

approved