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A001117
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a(n) = 3^n - 3*2^n + 3.
(Formerly M4219 N1763)
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24
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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
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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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]
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FORMULA
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a(n) = [n=0] + 3!*S(n, 3).
E.g.f.: 1 + (exp(x)-1)^3.
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
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MAPLE
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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
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.
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MATHEMATICA
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k=3; Prepend[Table[k!StirlingS2[n, k], {n, 1, 30}], 1] (* Robert A. Russell, Sep 25 2018 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Simpler description from Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
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STATUS
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approved
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