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 A005840 Expansion of (1-x)*e^x/(2-e^x). (Formerly M1872) 9
 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; text; internal format)
 OFFSET 0,3 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.] [Svante Janson, Apr 01 2009] If circuits were allowed that combine complex subcircuits in series or parallel, rather than requiring that one of them consists of a single resistor, then there are more additional possible resistances. For n = 4, there are additional 6 possible values. See illustration in links. - Kival Ngaokrajang, Aug 26 2013 (rephrased by Dave R.M. Langers, Nov 13 2013) Conjecture: A285868 (with offset 1) shows the associated connected threshold graphs. - R. J. Mathar, Apr 29 2019 Re: above conjecture - the number of connected threshold graphs on n labeled vertices is A317057 (see also A053525). [David Galvin, Oct 18 2021] REFERENCES Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 417. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.4(a). LINKS T. D. Noe, Table of n, a(n) for n=0..100 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, Apr 01 2009] Chao-Ping Chen and Xue-Feng Han, On Somos' quadratic recurrence constant, J. Number Theory, 166 (2016) 31-40. See page 34 equation (2.3). Mike de Vries, Graphical realizations of degree sequences, packing multiple colors and random graphs, Master's Thesis, Utrecht Univ. (Netherlands, 2023). Priyavrat Deshpande and Krishna Menon, Sketches, moves and partitions: counting regions of deformations of reflection arrangements, arXiv:2308.16653 [math.CO], 2023. P. Diaconis, S. Holmes and S. Janson, Threshold graph limits and random threshold graphs, Internet. math 5 (3) (2008) 267-320. D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021. Venkatesan Guruswami, Enumerative aspects of certain subclasses of perfect graphs, Discrete Math. 205 (1999), 97-117. Andrew H. Hoefel and Jeff Mermin, Gotzmann squarefree ideals, Ill. J. Math. 56, No. 2, 397-414 (2012), Proposition 3.13. Ricky I. Liu, K. Mészáros and A. H. Morales, Flow polytopes and the space of diagonal harmonics, arXiv preprint arXiv:1610.08370 [math.CO], 2016. Kival Ngaokrajang, Illustration for n = 4; [a1, a2, a3, a4] = [3, 5, 7, 9] Seunghyun Seo, The Catalan Threshold Arrangement, Journal of Integer Sequences, 2017 Vol. 20, #17.1.1. Sam Spiro, Counting Threshold Graphs with Eulerian Numbers, arXiv:1909.06518 [math.CO], 2019. 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. Eric Weisstein's World of Mathematics, Resistor Network FORMULA a(n) ~ n! * (1-log(2)) / (log(2))^(n+1). - Vaclav Kotesovec, Sep 29 2014 E.g.f.: (1 - x) * e^x / (2 - e^x). E.g.f. A(x) satisfies (1 - x) * A'(x) = A(x) * (A(x) - x). - Michael Somos, Aug 01 2016 a(n+1) = n*(a(n) - a(n-1)) + Sum_{k=0..n} binomial(n, k) * a(k) * a(n-k). - Michael Somos, Aug 01 2016 a(n) = (1-n) + Sum_{k=0..n-1} binomial(n, k) * a(k). - Michael Somos, Aug 01 2016 BINOMIAL transform of A053525. - Michael Somos, Aug 01 2016 a(n) = Sum_{k=1..n-1} (n-k)*A008292(n-1,k-1)*2^k, for n>=2. - Sam Spiro, Sep 22 2019 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, ... MAPLE A005840 := proc(n) option remember; 1 - n + add(binomial(n, k) * A005840(k), k = 0..n-1) end: seq(A005840(n), n = 0..20); # Peter Luschny, Oct 25 2021 MATHEMATICA nn = 20; Range[0, nn]! CoefficientList[Series[(1 - x) Exp[x]/(2 - Exp[x]), {x, 0, nn}], x] (* Harvey P. Dale, Jul 20 2011 *) PROG (PARI) my(x='x+O('x^30)); Vec(serlaplace((1-x)*exp(x)/(2-exp(x)))); \\ Michel Marcus, Jan 04 2016 CROSSREFS 2*A053525(n), n>1. Sequence in context: A337060 A141117 A145844 * A161881 A219358 A088791 Adjacent sequences: A005837 A005838 A005839 * A005841 A005842 A005843 KEYWORD nonn,easy,nice AUTHOR Simon Plouffe STATUS approved

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Last modified December 4 10:17 EST 2023. Contains 367560 sequences. (Running on oeis4.)