A000919 a(n) = 4^n - C(4,3)*3^n + C(4,2)*2^n - C(4,1).

0, 0, 0, 24, 240, 1560, 8400, 40824, 186480, 818520, 3498000, 14676024, 60780720, 249401880, 1016542800, 4123173624, 16664094960, 67171367640, 270232006800, 1085570781624, 4356217681200, 17466686971800, 69992221794000, 280345359228024, 1122510953731440

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Differences of 0: 4!*S(n,4).

Number of surjections from an n-element set onto a four-element set. - David Wasserman, Jun 06 2007

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

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.
• 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]
• 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.

Links

• T. D. Noe, Table of n, a(n) for n = 1..201
• K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters 50(1):20-22, January 2014.
• 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]
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Ethan Soloway, Megan Triplett, and Wenshi Zhao, On the Existence of Partition of the Hypercube Graph into 3 Initial Segments, arXiv:2501.05827 [math.CO], 2025. See p. 23.
• A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
• 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 (10,-35,50,-24).

Crossrefs

Cf. A001117, A001118.

Column 4 of A019538.

Programs

Maple

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
A000919:=24/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # Simon Plouffe in his 1992 dissertation

Mathematica

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 *)
k=4; Table[k!StirlingS2[n,k],{n,1,30}] (* Robert A. Russell, Sep 25 2018 *)

PARI

a(n) = 4!*stirling(n, 4, 2); \\ Altug Alkan, Sep 25 2018

Formula

G.f.: 24*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).

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

E.g.f.: (exp(x)-1)^4. - Geoffrey Critzer, Feb 11 2009

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

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