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A001118
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Number of labeled ordered set partitions into 5 parts for n>=1, a(0)=1.
(Formerly M5377 N2334)
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17
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1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 5103000, 29607600, 165528000, 901020120, 4809004200, 25292030400, 131542866000, 678330198120, 3474971465400, 17710714165200, 89904730860000, 454951508208120, 2296538629446600, 11570026582092000, 58200094019430000
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OFFSET
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0,6
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COMMENTS
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Previous name: Differences of 0; labeled ordered partitions into 5 parts.
Number of surjections from an n-element set onto a five-element set, with n >= 5. - Mohamed Bouhamida, 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
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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) = Sum_{i=0..4} (-1)^i*binomial(5, i)*(5-i)^n.
a(n) = [n=0] + 5!*S(n, 5).
E.g.f.: 1 + (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, 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
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MAPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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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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STATUS
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approved
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