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%I M4219 N1763 #98 Apr 02 2023 12:12:14
%S 1,0,0,6,36,150,540,1806,5796,18150,55980,171006,519156,1569750,
%T 4733820,14250606,42850116,128746950,386634060,1160688606,3483638676,
%U 10454061750,31368476700,94118013006,282379204836,847187946150,2541664501740,7625194831806
%N a(n) = 3^n - 3*2^n + 3.
%C Differences of 0. Labeled ordered partitions into 3 parts.
%C Number of surjections from an n-element set onto a three-element set, with n >= 3. - _Mohamed Bouhamida_, Dec 15 2007
%C 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|. - _Ross La Haye_, Mar 19 2009
%C For n>0, the number of rows of n colors using exactly three colors. For n=3, the six rows are ABC, ACB, BAC, BCA, CAB, and CBA. - _Robert A. Russell_, Sep 25 2018
%D 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 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
%D A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
%H T. D. Noe, <a href="/A001117/b001117.txt">Table of n, a(n) for n = 0..200</a>
%H John Elias, <a href="/A001117/a001117.png">Illustration of initial terms: Interior Sierpinski triangle</a>
%H K. S. Immink, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.299.4608">Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols</a>, 2013.
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.
%H P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H 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.
%H A. H. Voigt, <a href="/A000918/a000918.pdf">Theorie der Zahlenreihen und der Reihengleichungen</a>, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F a(n) = [n=0] + 3!*S(n, 3).
%F E.g.f.: 1 + (exp(x)-1)^3.
%F For n>=3: a(n+1) = 3*a(n) + 3*(2^n - 2) = 3*a(n) + 3*A000918(n). - _Geoffrey Critzer_, Feb 27 2009
%F G.f.: (-1-11*x^2+6*x)/((x-1)*(3*x-1)*(2*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
%p with(combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: 1, seq(count(subs(r=3,ZL),size=m),m=1..25); # _Zerinvary Lajos_, Mar 09 2007
%p A001117:=-6/(z-1)/(3*z-1)/(2*z-1); # Conjectured by _Simon Plouffe_ in his 1992 dissertation. Gives sequence except for three leading terms.
%t k=3; Prepend[Table[k!StirlingS2[n,k],{n,1,30}],1] (* _Robert A. Russell_, Sep 25 2018 *)
%o (PARI) a(n)=3^n-3*2^n+3 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000919, A001118.
%Y Column 3 of A019538 (n>0).
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_
%E Extended with formula and alternate description by _Christian G. Bower_, Aug 15 1998
%E Simpler description from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
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