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 A000919 a(n) = 4^n - C(4,3)*3^n + C(4,2)*2^n - C(4,1). (Formerly M5151 N2235) 14
 0, 0, 0, 24, 240, 1560, 8400, 40824, 186480, 818520, 3498000, 14676024, 60780720, 249401880, 1016542800, 4123173624, 16664094960, 67171367640, 270232006800, 1085570781624, 4356217681200, 17466686971800, 69992221794000, 280345359228024, 1122510953731440 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Differences of 0: 4!*S(n,4). Number of functions from an n-element set onto a four-element set. - David Wasserman, Jun 06 2007 Number of rows of n colors using exactly four colors.  For n=4, the 24 rows are the 24 permutations of ABCD. - 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. LINKS T. D. Noe, Table of n, a(n) for n = 1..201 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, Goschen, Leipzig, 1911, p. 31. 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 (10,-35,50,-24). FORMULA G.f.: 24x^3/((1-x)(1-2x)(1-3x)(1-4x)). a(n) = 4^n - binomial(4,3)*3^n + binomial(4,2)*2^n - binomial(4,1) = 24*A000453(n). - David Wasserman, Jun 06 2007 E.g.f.: (exp(x)-1)^4. - Geoffrey Critzer, Feb 11 2009 For n >= 4: a(n+1) = 4*a(n) + 4*(3^n - 3*2^n + 3) = 4*a(n) + 4*A001117(n). - Geoffrey Critzer, Feb 27 2009 a(n) = k!*S2(n,k), where k=4 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018 MAPLE with (combstruct):ZL:=[S, {S=Sequence(U, card=r), U=Set(Z, card>=1)}, labeled]: seq(count(subs(r=4, ZL), size=m), m=1..25); # Zerinvary Lajos, Mar 09 2007 A000919:=24/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation MATHEMATICA nn = 25; CoefficientList[Series[24 x^3/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, nn}], x] (* T. D. Noe, Jun 20 2012 *) k=4; Table[k!StirlingS2[n, k], {n, 1, 30}] (* Robert A. Russell, Sep 25 2018 *) PROG (PARI) a(n) = 4!*stirling(n, 4, 2); \\ Altug Alkan, Sep 25 2018 CROSSREFS Cf. A001117, A001118. Column 4 of A019538. Sequence in context: A253285 A052796 A056269 * A268966 A014340 A052753 Adjacent sequences:  A000916 A000917 A000918 * A000920 A000921 A000922 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)