cp's OEIS Frontend

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

%I A000920 M5473 N2370 #69 Aug 26 2024 04:38:38
%S A000920 0,0,0,0,0,720,15120,191520,1905120,16435440,129230640,953029440,
%T A000920 6711344640,45674188560,302899156560,1969147121760,12604139926560,
%U A000920 79694820748080,499018753280880,3100376804676480,19141689213218880,117579844328562000
%N A000920 Differences of 0: 6!*Stirling2(n,6).
%C A000920 Number of surjections from an n-element set onto a six-element set, with n >= 6. - _Mohamed Bouhamida_, Dec 15 2007
%C A000920 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 A000920 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 A000920 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
%D A000920 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000920 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000920 J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
%D A000920 A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
%H A000920 Vincenzo Librandi, <a href="/A000920/b000920.txt">Table of n, a(n) for n = 1..1000</a>
%H A000920 P. A. Piza, <a href="http://www.jstor.org/stable/3029339">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260.
%H A000920 P. A. Piza, <a href="/A001117/a001117.pdf">Kummer numbers</a>, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
%H A000920 A. H. Voigt, <a href="http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=05260001&amp;seq=7">Theorie der Zahlenreihen und der Reihengleichungen</a>, Leipzig, 1911.
%H A000920 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 A000920 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F A000920 a(n) = Sum((-1)^i*binomial(6, i)*(6-i)^n, i = 0 .. 5).
%F A000920 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 A000920 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 A000920 a(n) = 720*A000770(n). - _R. J. Mathar_, Apr 30 2015
%F A000920 E.g.f.: (exp(x) - 1)^6. - _Geoffrey Critzer_, May 17 2015
%p A000920 720/(-1+z)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1);
%t A000920 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 A000920 k=6; Table[k!StirlingS2[n,k],{n,1,30}] (* _Robert A. Russell_, Sep 25 2018 *)
%o A000920 (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 A000920 (PARI) a(n) = 6!*stirling(n, 6, 2); \\ _Altug Alkan_, Sep 25 2018
%Y A000920 Cf. A001117, A001118, A000918, A000919, A000770.
%Y A000920 Column 6 of A019538.
%K A000920 nonn,easy
%O A000920 1,6
%A A000920 _N. J. A. Sloane_