A000920 Differences of 0: 6!*Stirling2(n,6).

0, 0, 0, 0, 0, 720, 15120, 191520, 1905120, 16435440, 129230640, 953029440, 6711344640, 45674188560, 302899156560, 1969147121760, 12604139926560, 79694820748080, 499018753280880, 3100376804676480, 19141689213218880, 117579844328562000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of surjections from an n-element set onto a six-element set, with n >= 6. - Mohamed Bouhamida, Dec 15 2007

Number of rows of n colors using exactly six colors. For n=6, the 720 rows are the 720 permutations of ABCDEF. - 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 = 1..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]
• 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 (21,-175,735,-1624,1764,-720).

Crossrefs

Cf. A001117, A001118, A000918, A000919, A000770.

Column 6 of A019538.

Programs

Magma

[6^n-Binomial(6,5)*5^n+Binomial(6,4)*4^n-Binomial(6,3)*3^n+Binomial(6,2)*2^n-Binomial(6,1): n in [1..30]]; // Vincenzo Librandi, May 18 2015

Maple

720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);

Mathematica

CoefficientList[Series[(720*x^5)/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)),{x,0,30}],x] (* Vincenzo Librandi, Apr 11 2012 *)
k=6; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

PARI

a(n) = 6!*stirling(n, 6, 2); \\ Altug Alkan, Sep 25 2018

Formula

a(n) = Sum((-1)^i*binomial(6, i)*(6-i)^n, i = 0 .. 5).

a(n) = 6^n-C(6,5)*5^n+C(6,4)*4^n-C(6,3)*3^n+C(6,2)*2^n-C(6,1) with n>=6. - Mohamed Bouhamida, Dec 15 2007

G.f.: 720*x^6/((x-1)*(6*x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(5*x-1)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; checked and corrected by R. J. Mathar, Sep 16 2009]

a(n) = 720*A000770(n). - R. J. Mathar, Apr 30 2015

E.g.f.: (exp(x) - 1)^6. - Geoffrey Critzer, May 17 2015