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A060533
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Number of homeomorphically irreducible multigraphs (or series-reduced multigraphs or multigraphs without nodes of degree 2) on 3 labeled nodes.
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12
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1, 3, 0, 3, 9, 12, 19, 27, 36, 46, 57, 69, 82, 96, 111, 127, 144, 162, 181, 201, 222, 244, 267, 291, 316, 342, 369, 397, 426, 456, 487, 519, 552, 586, 621, 657, 694, 732, 771, 811, 852, 894, 937, 981, 1026, 1072, 1119, 1167, 1216, 1266, 1317, 1369, 1422
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OFFSET
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0,2
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REFERENCES
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I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
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LINKS
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FORMULA
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G.f.: (3*x^7 - 7*x^6 + 6*x^5 + 3*x^4 - 11*x^3 + 6*x^2 - 1)/(x - 1)^3.
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!.
a(n) = (1 + n)*(2 + n)/2 - 9 for n>4.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>7. (End)
Sum_{n>=3} 1/a(n) = 1/72 + 2*tan(sqrt(73)*Pi/2)*Pi/sqrt(73). - Amiram Eldar, Jan 08 2023
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MATHEMATICA
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PROG
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(PARI) Vec((3*x^7-7*x^6+6*x^5+3*x^4-11*x^3+6*x^2-1)/(x-1)^3 + O(x^60)) \\ Colin Barker, Nov 10 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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