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A007047
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Number of chains in power set of n-set.
(Formerly M2903)
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32
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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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history;
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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
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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V. Murali and B. B. Makamba, Finite Fuzzy Sets, International Journal of General Systems, Vol. 34, No. 1 (2005), pp. 61-75.
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FORMULA
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E.g.f.: exp(2*x)/(2-exp(x)).
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) = -(-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)):
# 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):
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MATHEMATICA
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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)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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