A000920 Differences of 0: 6!*Stirling2(n,6). (Formerly M5473 N2370)

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

Offset

1,6

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).

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

Maple

720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);
with (combstruct):ZL:=[S, {S=Sequence(U, card=r), U=Set(Z, card>=1)}, labeled]: seq(count(subs(r=6, ZL), size=m), m=1..22); # Zerinvary Lajos, Mar 09 2007

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 *)

Prog

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

Crossrefs

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

Column 6 of A019538.

Sequence in context: A112002 A004033 A056271 * A052779 A254079 A037212

Adjacent sequences: A000917 A000918 A000919 * A000921 A000922 A000923

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved