A001118 Number of labeled ordered set partitions into 5 parts for n>=1, a(0)=1. (Formerly M5377 N2334)

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

0,6

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).

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

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: A376375 A053567 A056270 * A375773 A373940 A354230

Adjacent sequences: A001115 A001116 A001117 * A001119 A001120 A001121

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Name edited by Harry Richman, Mar 31 2023

Status

approved