A000453 Stirling numbers of the second kind, S(n,4).

1, 10, 65, 350, 1701, 7770, 34105, 145750, 611501, 2532530, 10391745, 42355950, 171798901, 694337290, 2798806985, 11259666950, 45232115901, 181509070050, 727778623825, 2916342574750, 11681056634501, 46771289738810, 187226356946265, 749329038535350

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Given a set {1,2,3,4}, a(n) is the number of occurrences where the first 2 comes after the first '1', the first '3' after the first '2' and the first '4' after the first '3' in a list of n+3. For example, a(1): 1234; a(2): 11234, 12134, 12314, 12341, 12234, 12324, 12342, 12334, 12343, 12344. Related to the cereal box problem. - Kevin Nowaczyk, Aug 02 2007

a(n) is the number of partitions of [n] into 4 nonempty subsets. - Enrique Navarrete, Aug 27 2021

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
• 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).

Links

• T. D. Noe, Table of n, a(n) for n = 4..200
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• J. Brandts and C. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
• Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
• M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 347
• 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.
• Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Crossrefs

Cf. A008277 (Stirling2 triangle), A016269, A056280 (Mobius transform).

Programs

Maple

A000453:=1/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

t={}; Do[f=StirlingS2[n, 4]; AppendTo[t, f], {n, 120}]; t (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
LinearRecurrence[{10, -35, 50, -24}, {1, 10, 65, 350}, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

PARI

a(n)=(4^n-4*3^n+6*2^n-4)/24 \\ Charles R Greathouse IV, Sep 24 2015

Formula

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

E.g.f.: (exp(x)-1)^4/4!.

a(n) = (4^n - 4*3^n + 6*2^n - 4)/24. - Kevin Nowaczyk, Aug 02 2007

a(n) = det(|s(i+4,j+3)|, 1 <= i,j <= n-4), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4). - Wesley Ivan Hurt, Oct 10 2021