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A060534
Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 4 labeled nodes.
2
1, 6, 3, 10, 48, 84, 182, 372, 699, 1222, 2007, 3132, 4688, 6780, 9528, 13068, 17553, 23154, 30061, 38484, 48654, 60824, 75270, 92292, 112215, 135390, 162195, 193036, 228348, 268596, 314276, 365916, 424077, 489354, 562377, 643812, 734362, 834768, 945810
OFFSET
0,2
REFERENCES
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
FORMULA
G.f.: - (4*x^12 - 12*x^11 + 6*x^10 + 50*x^9 - 180*x^8 + 282*x^7 - 208*x^6 + 30*x^5 + 72*x^4 - 62*x^3 + 18*x^2 - 1)/((x - 1)^6).
E.g.f. for homeomorphically irreducible multigraphs with n nodes and k edges is (1 + x*y)^( - 1/2)*exp(x*y/2 + x^2*y^2/4)*Sum_{k >= 0} 1/(1 - x)^binomial(k, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!.
From Colin Barker, Nov 10 2016: (Start)
a(n) = 60 + 48*(1+n) - 12*(1+n)*(2+n) + (1+n)*(2+n)*(3+n)*(4+n)*(5+n)/120 for n>6.
a(n) = 6*a(n-1)- 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>12.
(End)
PROG
(PARI) Vec(-(4*x^12-12*x^11+6*x^10+50*x^9-180*x^8+282*x^7-208*x^6+30*x^5+72*x^4-62*x^3+18*x^2-1)/((x-1)^6) + O(x^40)) \\ Colin Barker, Nov 10 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Apr 01 2001
STATUS
approved