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 A001118 Differences of 0; labeled ordered partitions into 5 parts. (Formerly M5377 N2334) 14
 1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 5103000, 29607600, 165528000, 901020120, 4809004200, 25292030400, 131542866000, 678330198120, 3474971465400, 17710714165200, 89904730860000, 454951508208120, 2296538629446600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Number of surjections from an n-element set onto a five-element set, with n >= 5. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Dec 15 2007 For n > 0, the number of rows of n colors using exactly five colors. For n=5, the 120 rows are the 120 permutations of ABCDE. - 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 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 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] FORMULA a(n) = Sum_{i=0..4} (-1)^i*binomial(5, i)*(5-i)^n. a(n) = 5!*S(n, 5). E.g.f.: (e^x-1)^5. a(n) = 5^n - C(5,4)*4^n + C(5,3)*3^n - C(5,2)*2^n + C(5,1). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Dec 15 2007 G.f.: (-274*x^4 + 225*x^3 - 85*x^2 + 15*x - 1)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 MAPLE A001118:=-120/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # Conjectured (correctly) by Simon Plouffe in his 1992 dissertation. Gives sequence except for 5 leading terms. MATHEMATICA CoefficientList[Series[(-1-274*x^4+225*x^3-85*x^2+15*x)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 11 2012 *) k=5; Prepend[Table[k!StirlingS2[n, k], {n, 1, 30}], 1] (* Robert A. Russell, Sep 25 2018 *) PROG (PARI) a(n) = sum(i=0, 4, (-1)^i*binomial(5, i)*(5-i)^n); \\ Altug Alkan, Dec 04 2015 (PARI) Vec((-274*x^4 + 225*x^3 - 85*x^2 + 15*x - 1)/((x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1))+O(x^30)) \\ Stefano Spezia, Oct 16 2018 CROSSREFS Cf. A001117, A000919, A000920. Column 5 of A019538, n > 0. Sequence in context: A282899 A053567 A056270 * A052767 A110839 A219720 Adjacent sequences:  A001115 A001116 A001117 * A001119 A001120 A001121 KEYWORD nonn,easy AUTHOR EXTENSIONS Extended with formula and alternate description by Christian G. Bower, Aug 15 1998 STATUS approved

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Last modified January 23 00:56 EST 2019. Contains 319365 sequences. (Running on oeis4.)