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A005840
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Expansion of (1-x)*e^x/(2-e^x).
(Formerly M1872)
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5
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1, 1, 2, 8, 46, 332, 2874, 29024, 334982, 4349492, 62749906, 995818760, 17239953438, 323335939292, 6530652186218, 141326092842416, 3262247252671414, 80009274870905732, 2077721713464798210, 56952857434896699992, 1643312099715631960910
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also number of distinct resistances possible for n arbitrary resistors each connected in series or parallel with previous ones (cf. A051045).
The n-th term of A051045 uses the n different resistances 1, ..., n ohms, whereas the problem corresponding to A005840 allows arbitrary general resistances a1, a2, ..., an, chosen so as to give the maximum possible number of distinct equivalent resistances - Eric Weisstein..
Stanley's Problem 5.4(a) involves threshold graphs; Problem 5.4(c) involves hyperplane arrangements.
a(n) is the number of labeled threshold graphs on n vertices. [This is more specific than the reference to Stanley.] [From Svante Janson (svante.janson(AT)math.uu.se), Apr 01 2009]
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REFERENCES
| J.S. Beissinger and U.N. Peled, Enumeration of labelled threshold graphs and a theorem of Frobenius involving Eulerian polynomials, J Graphs Combin., 3 (1987), 213--219. MR903610 [From Svante Janson (svante.janson(AT)math.uu.se), Apr 01 2009]
Guruswami, Venkatesan, Enumerative aspects of certain subclasses of perfect graphs. Discrete Math. 205 (1999), 97-117.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, ``A zonotope associated with graphical degree sequences,'' in Applied Geometry and Discrete Combinatorics. DIMACS Series in Discrete Math., Amer. Math. Soc., Vol. 4, pp. 555-570, 1991.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.4(a).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
Eric Weisstein's World of Mathematics, Resistor Network
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EXAMPLE
| exp(x)*(1-x)/(2-exp(x)) = 1 + x + x^2 + 4/3*x^3 + 23/12*x^4 + 83/30*x^5 + 479/120*x^6 + 1814/315*x^7 + O(x^8); then the coefficients are multiplied by n! to get 1, 1, 2, 8, 46, 332, 2874, 29024, ...
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MATHEMATICA
| nn = 20; Range[0, nn]! CoefficientList[Series[(1 - x) Exp[x]/(2 - Exp[x]), {x, 0, nn}], x] (* From Harvey P. Dale, Jul 20 2011 *)
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CROSSREFS
| 2*A053525(n), n>1.
Sequence in context: A006664 A141117 A145844 * A161881 A088791 A111552
Adjacent sequences: A005837 A005838 A005839 * A005841 A005842 A005843
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KEYWORD
| nonn,easy,nice
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AUTHOR
| Simon Plouffe (simon.plouffe(AT)gmail.com)
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