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 A228696 Number of labeled graphs on n nodes with degree set {2,4}, with multiple edges and loops allowed. 2
 1, 2, 9, 65, 751, 13044, 320803, 10609256, 453774440, 24375801464, 1607240682376, 127684970262822, 12034618723574314, 1328262275098167080, 169754658940294717086, 24877923644434862091314, 4145248921431765036724198, 779383143209936088442516156, 164246020015613168238167009350 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)