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 A060577 Number of homeomorphically irreducible general graphs on 2 labeled nodes and with n edges. 1
 1, 1, 4, 6, 11, 17, 24, 32, 41, 51, 62, 74, 87, 101, 116, 132, 149, 167, 186, 206, 227, 249, 272, 296, 321, 347, 374, 402, 431, 461, 492, 524, 557, 591, 626, 662, 699, 737, 776, 816, 857, 899, 942, 986, 1031, 1077, 1124, 1172, 1221, 1271, 1322, 1374, 1427 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A homeomorphically irreducible general graph is a graph with multiple edges and loops and without nodes of degree 2. REFERENCES I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983. LINKS Table of n, a(n) for n=0..52. V. Jovovic, Generating functions for homeomorphically irreducible general graphs on n labeled nodes V. Jovovic, Recurrences for the numbers of homeomorphically irreducible general graphs on m labeled nodes and n edges Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA G.f.: (2*x^5 - 4*x^4 + 4*x^3 - 4*x^2 + 2*x - 1)/(x - 1)^3. E.g.f. for homeomorphically irreducible general graphs 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 + 1, 2)*exp( - x^2*y*k^2/(2*(1 + x*y)) - x^2*y*k/2)*y^k/k!. From Marco Ripà, Aug 20 2015: (Start) a(n) = ceiling( (1/2)*(3*n^2 - 10*n + 9)/(n - 2) ) + floor( (3/2)*(n-1)^2 ) - n^2 + 3*n - 3 with n > 2, a(0) = a(1) = 1, a(2) = 4. a(n) = n*(n + 3)/2 - 3 for n > 2. a(n) = A046691(n-1) for n > 2. (End) MAPLE gf := (2*x^5 - 4*x^4 + 4*x^3 - 4*x^2 + 2*x - 1)/(x - 1)^3: s := series(gf, x, 100): for i from 0 to 100 do printf(`%d, `, coeff(s, x, i)) od: MATHEMATICA Join[{1, 1, 4}, Table[n (n + 3)/2 - 3, {n, 3, 60}]] (* Bruno Berselli, Aug 20 2015 *) CROSSREFS Cf. A003514, A046691, A060516, A060533-A060537, A060576-A060581. Sequence in context: A343345 A343346 A296468 * A360042 A360361 A197985 Adjacent sequences: A060574 A060575 A060576 * A060578 A060579 A060580 KEYWORD nonn,easy AUTHOR Vladeta Jovovic, Apr 04 2001 EXTENSIONS More terms from James A. Sellers, Apr 04 2001 STATUS approved

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Last modified March 1 08:46 EST 2024. Contains 370430 sequences. (Running on oeis4.)