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%I M5151 N2235 #81 May 29 2023 20:58:06
%S 0,0,0,24,240,1560,8400,40824,186480,818520,3498000,14676024,60780720,
%T 249401880,1016542800,4123173624,16664094960,67171367640,270232006800,
%U 1085570781624,4356217681200,17466686971800,69992221794000,280345359228024,1122510953731440
%N a(n) = 4^n - C(4,3)*3^n + C(4,2)*2^n - C(4,1).
%C Differences of 0: 4!*S(n,4).
%C Number of surjections from an n-element set onto a four-element set. - _David Wasserman_, Jun 06 2007
%C Number of rows of n colors using exactly four colors. For n=4, the 24 rows are the 24 permutations of ABCD. - _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 K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, http://www.exp-math.uni-essen.de/~immink/pdf/jsac13.pdf, 2013. [This link no longer works, but please do not delete this reference, for historical reasons. _Michel Marcus_ has suggested that the Immink link below points to the published version of the original reference, and I agree. - _N. J. A. Sloane_, May 29 2023]
%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.
%H T. D. Noe, <a href="/A000919/b000919.txt">Table of n, a(n) for n = 1..201</a>
%H K. S. Immink, <a href="http://dx.doi.org/10.1049/el.2013.3558">Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols</a>, Electronics Letters 50(1):20-22, January 2014.
%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 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%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>, Goschen, Leipzig, 1911, p. 31.
%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_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F G.f.: 24*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).
%F a(n) = 4^n - binomial(4,3)*3^n + binomial(4,2)*2^n - binomial(4,1) = 24*A000453(n). - _David Wasserman_, Jun 06 2007
%F E.g.f.: (exp(x)-1)^4. - _Geoffrey Critzer_, Feb 11 2009
%F For n >= 4: a(n+1) = 4*a(n) + 4*(3^n - 3*2^n + 3) = 4*a(n) + 4*A001117(n). - _Geoffrey Critzer_, Feb 27 2009
%F a(n) = k!*S2(n,k), where k=4 is the number of colors and S2 is the Stirling subset number. - _Robert A. Russell_, Sep 25 2018
%p with (combstruct):ZL:=[S,{S=Sequence(U,card=r),U=Set(Z,card>=1)}, labeled]: seq(count(subs(r=4,ZL),size=m),m=1..25); # _Zerinvary Lajos_, Mar 09 2007
%p A000919:=24/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # _Simon Plouffe_ in his 1992 dissertation
%t nn = 25; CoefficientList[Series[24 x^3/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, nn}], x] (* _T. D. Noe_, Jun 20 2012 *)
%t k=4; Table[k!StirlingS2[n,k],{n,1,30}] (* _Robert A. Russell_, Sep 25 2018 *)
%o (PARI) a(n) = 4!*stirling(n, 4, 2); \\ _Altug Alkan_, Sep 25 2018
%Y Cf. A001117, A001118.
%Y Column 4 of A019538.
%K nonn,easy
%O 1,4
%A _N. J. A. Sloane_
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