0,2

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

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 04 2009: (Start)

Equals binomial transform of A000629: (1, 2, 6, 26, 150, 1082,...)

and double binomial transform of A000670: (1, 1, 3, 13, 75, 541,...). (End)

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

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

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

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

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

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

Index entries for sequences related to posets

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 (benoit7848c(AT)orange.fr), 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

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

Cf. A000629.

A000629, A000670 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 04 2009]

Sequence in context: A020043 A113712 A056199 this_sequence A129097 A014510 A058799

Adjacent sequences: A007044 A007045 A007046 this_sequence A007048 A007049 A007050

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com), Roger B. Nelsen