A000481 Stirling numbers of the second kind, S(n,5).

1, 15, 140, 1050, 6951, 42525, 246730, 1379400, 7508501, 40075035, 210766920, 1096190550, 5652751651, 28958095545, 147589284710, 749206090500, 3791262568401, 19137821912055, 96416888184100, 485000783495250, 2436684974110751, 12230196160292565, 61338207158409090

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn

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=5..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].
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 348
• Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
• 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

Crossrefs

Cf. A008277.

Programs

Maple

A000481:=-1/(z-1)/(4*z-1)/(-1+3*z)/(2*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation
a := n -> (1-4^n+2*(3^n-2^n)+5^(n-1))/24:
seq(a(n), n=5..29); # Peter Luschny, May 09 2015

Mathematica

lst={};Do[f=StirlingS2[n, 5];AppendTo[lst, f], {n, 5, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x)), {x, 0, 25}], x]  (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
StirlingS2[Range[5,30],5] (* Harvey P. Dale, May 15 2017 *)

Formula

a(n) = A008277(n, 5) (Stirling2 triangle).

G.f.: x^5/product(1-k*x, k=1..5).

E.g.f.: ((exp(x)-1)^5)/5!.

a(n) = sum(sum(binomial(k,r)*(15)^(k-r)*sum((-85)^(r-m)*binomial(r,m)*sum(binomial(m,j)*binomial(j,n-m-k-j-r)*(225)^(m-j)*(-274)^(r+m+k+2*j-n)*(120)^(n-m-k-j-r),j,0,m),m,0,r),r,0,k),k,1,n), n>0. - Vladimir Kruchinin, Aug 30 2010

a(n) = det(|s(i+5,j+4)|, 1 <= i,j <= n-5), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013

Extensions

More terms from Sean A. Irvine, Nov 14 2010