cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

1, 1, 2, 8, 46, 332, 2874, 29024, 334982, 4349492, 62749906, 995818760, 17239953438, 323335939292, 6530652186218, 141326092842416, 3262247252671414, 80009274870905732, 2077721713464798210, 56952857434896699992, 1643312099715631960910
Offset: 0

Views

Author

Simon Plouffe

Keywords

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]

Examples

			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, ...

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

Crossrefs

2*A053525(n), n>1.

Programs

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

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

A285867 Triangle T(n, k) read by rows: T(n, k) = S2(n, k)*k! + S2(n, k-1)*(k-1)! with the Stirling2 triangle S2 = A048993.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 7, 12, 0, 1, 15, 50, 60, 0, 1, 31, 180, 390, 360, 0, 1, 63, 602, 2100, 3360, 2520, 0, 1, 127, 1932, 10206, 25200, 31920, 20160, 0, 1, 255, 6050, 46620, 166824, 317520, 332640, 181440, 0, 1, 511, 18660, 204630, 1020600, 2739240, 4233600, 3780000, 1814400, 0, 1, 1023, 57002, 874500, 5921520, 21538440, 46070640, 59875200, 46569600, 19958400
Offset: 0

Views

Author

Wolfdieter Lang, May 03 2017

Keywords

Comments

This triangle T(n, k) appears in the e.g.f. of the sum of powers SP(n, m) = Sum_{j=0..m} j^n, n >= 0, m >= 0 with 0^0:=1 as ESP(n, t) = exp(t)*(Sum_{k=0..n} T(n, k)*t^k/k! + t^(n+1)/(n+1)), n >= 0.
The sub-triangle T(n, k) for 1 <= k <=n, see A028246(n+1,k) (diagonal not needed).
For S2(n, m)*m! see A131689.
The columns (starting sometimes with n=k) are A000007, A000012, A000225, A028243(n-1), A028244(n-1), A028245(n-1), A032180(n-1), A228909, A228910, A228911, A228912, A228913. See below for the e.g.f.s and o.g.f.s.
The row sums are 1 for n=1 and A000629(n) - n! for n >= 1, See A285868.

Examples

			The triangle T(n, k) begins:
n\k 0  1    2     3      4       5        6        7        8        9  ...
0:  1
1:  0  1
2:  0  1    3
3:  0  1    7    12
4:  0  1   15    50     60
5:  0  1   31   180    390     360
6:  0  1   63   602   2100    3360     2520
7:  0  1  127  1932  10206   25200    31920    20160
8:  0  1  255  6050  46620  166824   317520   332640   181440
9:  0  1  511 18660 204630 1020600  2739240  4233600  3780000  1814400
...

Crossrefs

Cf. A000629, A028246, A048993, A131689, A285868.

Programs

  • Mathematica
    Table[If[k == 0, Boole[n == 0], StirlingS2[n, k] k! + StirlingS2[n, k - 1] (k - 1)!], {n, 0, 10}, {k, 0, n}] (* Michael De Vlieger, May 08 2017 *)

Formula

T(n, k) = A131689(n, k) + A131689(n, k-1), 0 <= k <= n, with A131689(n, -1) = 0.
T(0, 0) = 1 and T(n, k) = Stirling2(n+1, k)*(k-1)! for n >= k >= 1. For Stirling2 see A048993. Stirling2(n, k)*(k-1)! = A028246(n, k) for n >= k >= 1.
Recurrence: T(0, 0) = 1, T(n, n) = (n+1)!/2, T(n, -1) = 0, T(n, k) = 0 if n < k, and T(n, k) = (k-1)*T(n-1, k-1) + k*T(n-1, k), for n > k >= 0.
E.g.f. for column k=0 is 1, and for k >= 1: Sum_{j=1..k}((-1)^(k-j) * binomial(k-1, j-1) * exp(j*x)) - x^(k-1).
O.g.f. for column k = 0 is 1, and for k >= 1: ((k-1)!*x^(k-1) / Product_{j=1..k} (1-j*x)) - (k-1)!*x^(k-1).
Showing 1-2 of 2 results.

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