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A007047
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Number of chains in power set of n-set.
(Formerly M2903)
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30
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1, 3, 11, 51, 299, 2163, 18731, 189171, 2183339, 28349043, 408990251, 6490530291, 112366270379, 2107433393523, 42565371881771, 921132763911411, 21262618727925419, 521483068116543603, 13542138653027381291, 371206349277313644531
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OFFSET
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0,2
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COMMENTS
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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 the binomial transform of A000629: (1, 2, 6, 26, 150, 1082, ...) and the 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
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REFERENCES
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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).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..424 (first 101 terms from T. D. Noe)
P. Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Lisa Berry, Stefan Forcey, Maria Ronco, and Patrick Showers, Species substitution, graph suspension, and graded hopf algebras of painted tree polytopes.
Jun Ma, S.-M. Ma, and Y.-N. Yeh, Recurrence relations for binomial-Eulerian polynomials, arXiv preprint arXiv:1711.09016 [math.CO], 2017.
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015-2016.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
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, No. 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 and 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
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FORMULA
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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 1*k_1 + 2*k_2 + ... + n*k_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
a(n) = 4 * A000670(n) - 1 for n > 0. - Alois P. Heinz, Feb 07 2020
a(n) = -(-1)^n Phi(2,-n,-1), where Phi(z,s,a) is the Lerch Zeta function. - Federico Provvedi, Sep 05 2020
a(n) = 1 + 2 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n,k) * a(k). - Ilya Gutkovskiy, Apr 28 2021
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MAPLE
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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
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 4,
add(b(n-j)*binomial(n, j), j=1..n))
end:
a:= n-> `if`(n=0, 1, b(n)-1):
seq(a(n), n=0..21); # Alois P. Heinz, Feb 07 2020
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MATHEMATICA
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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-François 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 *)
Table[-(-1)^k HurwitzLerchPhi[2, -k, -1], {k, 0, 30}] (* Federico Provvedi, Sep 05 2020 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane, Roger B. Nelsen
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STATUS
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approved
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