%I A001117 M4219 N1763
%S A001117 1,0,0,6,36,150,540,1806,5796,18150,55980,171006,519156,1569750,4733820,
%T A001117 14250606,42850116,128746950,386634060,1160688606,3483638676,10454061750,
%U A001117 31368476700,94118013006,282379204836,847187946150,2541664501740,7625194831806
%N A001117 3^n-3*2^n+3.
%C A001117 Differences of 0. Labeled ordered partitions into 3 parts.
%C A001117 Number of surjections from an n-element set onto a three-element set, 
               with n >= 3. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Dec 15 2007
%C A001117 Let P(A) be the power set of an n-element set A and R be a relation on 
               P(A) such that for all x, y of P(A), xRy if either 0) x is not a 
               subset of y and y is not a subset of x and x and y are disjoint, 
               or 1) x is a proper subset of y or y is a proper subset of x and 
               x and y are intersecting. Then a(n+1) = |R|. [From Ross La Haye (rlahaye(AT)new.rr.com), 
               Mar 19 2009]
%D A001117 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001117 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001117 H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd 
               ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity 
               Univ., San Antonio, TX, Vol. 2, p. 212.
%D A001117 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               33.
%D A001117 J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
%D A001117 A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, 
               Leipzig, 1911, p. 31.
%D A001117 Ross La Haye, Binary Relations on the Power Set of an n-Element Set, 
               Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From 
               Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 2009]
%H A001117 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001117 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001117 A. H. Voigt, <a href="http://historical.library.cornell.edu/cgi-bin/cul.math/
               docviewer?did=05260001&seq=7">Theorie der Zahlenreihen und der Reihengleichungen</
               a>, Leipzig, 1911.
%F A001117 3!*S(n, 3). E.g.f.: (e^x-1)^3.
%F A001117 For n>=3: a(n+1)=3*a(n)+3*[(2^n)-2]=3*a(n)+3*A000918(n) [From Geoffrey 
               Critzer (critzer.geoffrey(AT)usd443.org), Feb 27 2009]
%F A001117 G.f.:(-1-11*x^2+6*x)/((x-1)*(3*x-1)*(2*x-1)) [From Maksym Voznyy (voznyy(AT)mail.ru), 
               Jul 26 2009]
%p A001117 with (combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: 
               seq(count(subs(r=3,ZL),size=m),m=0..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 09 2007
%p A001117 A001117:=-6/(z-1)/(3*z-1)/(2*z-1); [Conjectured by S. Plouffe in his 
               1992 dissertation. Gives sequence except for three leading terms.]
%Y A001117 Cf. A000919, A001118, A019538.
%Y A001117 Sequence in context: A056375 A018214 A056268 this_sequence A055404 A132165 
               A074444
%Y A001117 Adjacent sequences: A001114 A001115 A001116 this_sequence A001118 A001119 
               A001120
%K A001117 nonn,easy
%O A001117 0,4
%A A001117 N. J. A. Sloane (njas(AT)research.att.com).
%E A001117 Extended with formula and alternate description by Christian G. Bower 
               (bowerc(AT)usa.net), Aug 15 1998.
%E A001117 Simpler description from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), 
               Apr 07 2001