A051045 -id:A051045 - cp's OEIS frontend

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

1, 1, 2, 8, 46, 332, 2874, 29024, 334982, 4349492, 62749906, 995818760, 17239953438, 323335939292, 6530652186218, 141326092842416, 3262247252671414, 80009274870905732, 2077721713464798210, 56952857434896699992, 1643312099715631960910

Offset: 0

Simon Plouffe

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]

			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, ...

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).

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

2*A053525(n), n>1.

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

nn = 20; Range[0, nn]! CoefficientList[Series[(1 - x) Exp[x]/(2 - Exp[x]), {x, 0, nn}], x] (* Harvey P. Dale, Jul 20 2011 *)

my(x='x+O('x^30)); Vec(serlaplace((1-x)*exp(x)/(2-exp(x)))); \\ Michel Marcus, Jan 04 2016

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

A123750 Number of distinct resistances possible with at most n arbitrary resistors connected in series or in parallel.

Original entry on oeis.org

1, 4, 17, 94, 667, 5752, 58053, 669970, 8698991, 125499820, 1991637529, 34479906886, 646671878595, 13061304372448, 282652185684845, 6524494505342842, 160018549741811479, 4155443426929596436, 113905714869793400001, 3286624199431263921838

Offset: 1

I. N. Galidakis (jgal(AT)ath.forthnet.gr), Nov 28 2006

The difference between this problem and A005840 and A051045 is the word "at most". In this problem, at most n different resistors are used to generate all possible resistances using in series and in parallel wirings, also including resistances where some of the resistors from the collection 1,2,...,n, are not used.

Cf. A005840, A051045.

a:= n-> n!* coeff(series(exp(x)*(-2*exp(x) +
            exp(x)*x + 2)/(-2 + exp(x)), x, n+1),x,n):
seq(a(n), n=1..25);

a(n) = 2 * A005840(n) + n - 2, n > 1.

E.g.f.: exp(x)*(-2*exp(x) + exp(x)*x + 2)/(-2 + exp(x)).

A232005 Number of distinct resistances that can be produced from a circuit of resistors with resistances 1, 2, ..., n using only series and parallel combinations.

Original entry on oeis.org

1, 2, 8, 48, 386, 3781, 49475, 762869, 13554897, 266817541

Offset: 1

Dave R.M. Langers, Nov 16 2013

Found by exhaustive search: all configurations of resistors were enumerated, resistances calculated, sorted, and distinct values counted.
This sequence allows any circuits to be combined in series or in parallel (akin A000084); A051045 requires circuits to be combined with a single resistor at a time.
This sequence regards circuits as distinct only if their resistance is different; A006351 regards circuits distinct if their configuration is different, although some may have the same resistance.
This sequence considers resistors with contiguous resistances 1, 2, ..., n; A005840 considers arbitrarily different resistors, while A048211 considers n equal resistances.

			a(2) = 2 since given a 1-ohm and a 2-ohm resistor, a series circuit yields 3 ohms, while a parallel circuit yields 2/3 ohms, which thus yields two distinct resistances.

Cf. A005840, A006352, A048211, A051045.

cp's OEIS Frontend

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

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A123750 Number of distinct resistances possible with at most n arbitrary resistors connected in series or in parallel.

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A232005 Number of distinct resistances that can be produced from a circuit of resistors with resistances 1, 2, ..., n using only series and parallel combinations.

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