

A005840


Expansion of (1x)*e^x/(2e^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
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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 nth 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



FORMULA

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(n1)) + Sum_{k=0..n} binomial(n, k) * a(k) * a(nk).  Michael Somos, Aug 01 2016
a(n) = (1n) + Sum_{k=0..n1} binomial(n, k) * a(k).  Michael Somos, Aug 01 2016
a(n) = Sum_{k=1..n1} (nk)*A008292(n1,k1)*2^k, for n>=2.  Sam Spiro, Sep 22 2019


EXAMPLE

exp(x)*(1x)/(2exp(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..n1) end:


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((1x)*exp(x)/(2exp(x)))); \\ Michel Marcus, Jan 04 2016


CROSSREFS



KEYWORD

nonn,easy,nice


AUTHOR



STATUS

approved



