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A001118 Number of labeled ordered set partitions into 5 parts for n>=1, a(0)=1.

%I A001118 M5377 N2334 #81 Jun 25 2023 02:22:39
%S A001118 1,0,0,0,0,120,1800,16800,126000,834120,5103000,29607600,165528000,
%T A001118 901020120,4809004200,25292030400,131542866000,678330198120,
%U A001118 3474971465400,17710714165200,89904730860000,454951508208120,2296538629446600,11570026582092000,58200094019430000
%N A001118 Number of labeled ordered set partitions into 5 parts for n>=1, a(0)=1.
%C A001118 Previous name: Differences of 0; labeled ordered partitions into 5 parts.
%C A001118 Number of surjections from an n-element set onto a five-element set, with n >= 5. - _Mohamed Bouhamida_, Dec 15 2007
%C A001118 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
%D A001118 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.
%D A001118 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
%D A001118 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001118 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001118 J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
%D A001118 A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
%H A001118 Vincenzo Librandi, <a href="/A001118/b001118.txt">Table of n, a(n) for n = 0..1000</a>
%H A001118 K. S. Immink, <a href="http://www.exp-math.uni-essen.de/~immink/pdf/jsac13.pdf">Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols</a>, 2013.
%H A001118 P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.
%H A001118 P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
%H A001118 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A001118 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001118 A. H. Voigt, <a href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05260001&amp;seq=7">Theorie der Zahlenreihen und der Reihengleichungen</a>, Leipzig, 1911.
%H A001118 A. H. Voigt, <a href="/A000918/a000918.pdf">Theorie der Zahlenreihen und der Reihengleichungen</a>, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
%H A001118 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15, -85, 225, -274, 120).
%F A001118 a(n) = Sum_{i=0..4} (-1)^i*binomial(5, i)*(5-i)^n.
%F A001118 a(n) = [n=0] + 5!*S(n, 5).
%F A001118 E.g.f.: 1 + (e^x-1)^5.
%F A001118 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
%F A001118 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
%p A001118 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.
%t A001118 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 *)
%t A001118 k=5; Prepend[Table[k!StirlingS2[n,k],{n,1,30}],1] (* _Robert A. Russell_, Sep 25 2018 *)
%o A001118 (PARI) a(n) = sum(i=0, 4, (-1)^i*binomial(5, i)*(5-i)^n); \\ _Altug Alkan_, Dec 04 2015
%o A001118 (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
%Y A001118 Cf. A001117, A000919, A000920.
%Y A001118 Column 5 of A019538, n > 0.
%K A001118 nonn,easy
%O A001118 0,6
%A A001118 _N. J. A. Sloane_
%E A001118 Extended with formula and alternate description by _Christian G. Bower_, Aug 15 1998
%E A001118 Name edited by _Harry Richman_, Mar 31 2023