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%I M5473 N2370 #63 Oct 21 2022 22:08:38
%S 0,0,0,0,0,720,15120,191520,1905120,16435440,129230640,953029440,
%T 6711344640,45674188560,302899156560,1969147121760,12604139926560,
%U 79694820748080,499018753280880,3100376804676480,19141689213218880,117579844328562000
%N Differences of 0: 6!*Stirling2(n,6).
%C Number of surjections from an n-element set onto a six-element set, with n >= 6. - _Mohamed Bouhamida_, Dec 15 2007
%C 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
%D 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 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
%D A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
%H Vincenzo Librandi, <a href="/A000920/b000920.txt">Table of n, a(n) for n = 1..1000</a>
%H P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.
%H P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
%H A. H. Voigt, <a href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05260001&seq=7">Theorie der Zahlenreihen und der Reihengleichungen</a>, Leipzig, 1911.
%H 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 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F a(n) = Sum((-1)^i*binomial(6, i)*(6-i)^n, i = 0 .. 5).
%F 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
%F 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]
%F a(n) = 720*A000770(n). - _R. J. Mathar_, Apr 30 2015
%F E.g.f.: (exp(x) - 1)^6. - _Geoffrey Critzer_, May 17 2015
%p 720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);
%p 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
%t 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 *)
%t k=6; Table[k!StirlingS2[n,k],{n,1,30}] (* _Robert A. Russell_, Sep 25 2018 *)
%o (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
%o (PARI) a(n) = 6!*stirling(n, 6, 2); \\ _Altug Alkan_, Sep 25 2018
%Y Cf. A001117, A001118, A000918, A000919, A000770.
%Y Column 6 of A019538.
%K nonn,easy
%O 1,6
%A _N. J. A. Sloane_
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