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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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
• K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, 2013.
• 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; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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]
• Index entries for linear recurrences with constant coefficients, signature (15, -85, 225, -274, 120).

Crossrefs

Cf. A001117, A000919, A000920.

Column 5 of A019538, n > 0.

Programs

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 *)

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

Formula

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

Extensions

Extended with formula and alternate description by Christian G. Bower, Aug 15 1998

Name edited by Harry Richman, Mar 31 2023