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A007047 Number of chains in power set of n-set.
(Formerly M2903)
18
1, 3, 11, 51, 299, 2163, 18731, 189171, 2183339, 28349043, 408990251, 6490530291, 112366270379, 2107433393523, 42565371881771, 921132763911411, 21262618727925419, 521483068116543603, 13542138653027381291, 371206349277313644531 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Stirling transform of A052849(n-1)=[1,2,4,12,48,...] is a(n-1)=[1,3,11,51,299,...]. - Michael Somos, Mar 04 2004

It is interesting to note that a chain in the power set of a set X can be thought of as a fuzzy subset of X and conversely. Chains originating with empty set are fuzzy subsets with empty core and those chains not ending with the whole set are with support strictly contained in X. - Venkat Murali (v.murali(AT)ru.ac.za), May 18 2005

Equals binomial transform of A000629: (1, 2, 6, 26, 150, 1082,...) and double binomial transform of A000670: (1, 1, 3, 13, 75, 541,...). - Gary W. Adamson, Aug 04 2009

Row sums of A038719. - Peter Bala, Jul 09 2014

Also the number of restricted barred preferential arrangements of an n-set having two bars, where one fixed section is a free section and the other two sections are restricted sections. - Sithembele Nkonkobe, Jun 16 2015

REFERENCES

Jun Ma, SM Ma, YN Yeh, Recurrence relations for binomial-Eulerian polynomials, arXiv preprint arXiv:1711.09016, 2017

V. Murali, Counting fuzzy subsets of a finite set, preprint, Rhodes University, Grahamstown 6140, South Africa, 2003.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

P. Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.

T. Manneville, V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152, 2015

V. Murali, Number of fuzzy subsets of a finite set, fuzzy systems research group, Universities of Rhodes and Fort Hare. [broken link]

V. Murali and B. B. Makamba, Finite Fuzzy Sets, International Journal of General Systems, Vol. 34 (1) (2005), pp. 61-75.

R. B. Nelsen, Letter to N. J. A. Sloane, Aug. 1991

R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.

S. Nkonkobe, V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO] Apr 2015

Index entries for sequences related to posets

FORMULA

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

a(n) = sum(k>=1, (k+1)^n/2^k) = 2*A000629(n)-1. - Benoit Cloitre, Sep 08 2002

a(n) = one less than sum of quotients with numerator 4 times (n!)((k_1 + k_2 +...+ k_n)!) and with denominator (k_1!k_2!...k_n!)(1!^k_1 2!^k_2...n!^k_n) where the sum is taken over all partitions 1k_1 + 2k_2 + ...+ nk_n = n. E.g. a(1) = 3 because the membership value of x to {x} is either 1, alpha with 0 < alpha < 1 or 0. a(2) = 11 since the membership values x and y to {x, y} are 1>= alpha >= beta >= 0 for {empty set, x, y} in that order or {empty set, y, x} exercising all possible > or =. - Venkat Murali (v.murali(AT)ru.ac.za), May 18 2005

G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(k+1) - 2*x^2*(k+1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2013

a(n) ~ 2*n! / (log(2))^(n+1). - Vaclav Kotesovec, Sep 27 2017

MAPLE

P := proc(n, x) option remember; if n = 0 then 1 else

   (n*x+2*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x);

   expand(%) fi end:

A007047 := n -> 2^n*subs(x=1/2, P(n, x)):

seq(A007047(n), n=0..19);  # Peter Luschny, Mar 07 2014

MATHEMATICA

Table[LerchPhi[1/2, -n, 2]/2, {n, 0, 10}] (* Vladimir Reshetnikov, Feb 16 2011 *)

Table[2*PolyLog[-n, 1/2] - 1 , {n, 0, 19}] (* Jean-Fran├žois Alcover, Aug 14 2013 *)

With[{nn=20}, CoefficientList[Series[Exp[2x]/(2-Exp[x]), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Dec 08 2015 *)

PROG

(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst((y+1)^2/(1-y), y, exp(x+x*O(x^n))-1), n));

(PARI) x='x+O('x^66); Vec(serlaplace(exp(2*x)/(2-exp(x)))) \\ Joerg Arndt, Aug 14 2013

(Haskell)

a007047 = sum . a038719_row  -- Reinhard Zumkeller, Feb 05 2014

CROSSREFS

Cf. A000629, A000629, A000670. Row sums of A038719.

Sequence in context: A113712 A056199 A230008 * A182176 A244754 A129097

Adjacent sequences:  A007044 A007045 A007046 * A007048 A007049 A007050

KEYWORD

nonn,nice,easy,changed

AUTHOR

N. J. A. Sloane, Roger B. Nelsen

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)