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A005840 Expansion of (1-x)*e^x/(2-e^x).
(Formerly M1872)
6
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)

REFERENCES

Miklos Bona, 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)

Venkatesan Guruswami, Enumerative aspects of certain subclasses of perfect graphs, Discrete Math. 205 (1999), 97-117.

Kival Ngaokrajang, Illustration for n = 4; [a1, a2, a3, a4] = [3, 5, 7, 9]

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

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, ...

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) 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: A276358 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 February 22 15:12 EST 2017. Contains 282487 sequences.