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 A001117 a(n) = 3^n - 3*2^n + 3. (Formerly M4219 N1763) 20
 1, 0, 0, 6, 36, 150, 540, 1806, 5796, 18150, 55980, 171006, 519156, 1569750, 4733820, 14250606, 42850116, 128746950, 386634060, 1160688606, 3483638676, 10454061750, 31368476700, 94118013006, 282379204836, 847187946150, 2541664501740, 7625194831806 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Differences of 0. Labeled ordered partitions into 3 parts. Number of surjections from an n-element set onto a three-element set, with n >= 3. - Mohamed Bouhamida, Dec 15 2007 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 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 REFERENCES 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. J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54. A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31. LINKS T. D. Noe, Table of n, a(n) for n = 0..200 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. P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy] Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Leipzig, 1911. A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only] Index entries for linear recurrences with constant coefficients, signature (6,-11,6). FORMULA a(n) = 3!*S(n, 3). E.g.f.: (exp(x)-1)^3. For n>=3: a(n+1) = 3*a(n)+3*[(2^n)-2] = 3*a(n)+3*A000918(n). - Geoffrey Critzer, Feb 27 2009 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] MAPLE 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, Mar 09 2007 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. MATHEMATICA k=3; Prepend[Table[k!StirlingS2[n, k], {n, 1, 30}], 1] (* Robert A. Russell, Sep 25 2018 *) PROG (PARI) a(n)=3^n-3*2^n+3 \\ Charles R Greathouse IV, Sep 24 2015 CROSSREFS Cf. A000919, A001118. Column 3 of A019538 (n>0). Sequence in context: A223841 A210322 A056268 * A224149 A055404 A223946 Adjacent sequences:  A001114 A001115 A001116 * A001118 A001119 A001120 KEYWORD nonn,easy AUTHOR EXTENSIONS Extended with formula and alternate description by Christian G. Bower, Aug 15 1998 Simpler description from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001 STATUS approved

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Last modified August 12 05:33 EDT 2020. Contains 336438 sequences. (Running on oeis4.)