A000770 Stirling numbers of the second kind, S(n,6).

1, 21, 266, 2646, 22827, 179487, 1323652, 9321312, 63436373, 420693273, 2734926558, 17505749898, 110687251039, 693081601779, 4306078895384, 26585679462804, 163305339345225, 998969857983405, 6090236036084530, 37026417000002430, 224595186974125331

Offset: 6

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=6..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 349
• 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

a(n)= A008277(n, 6) (Stirling2 triangle).

Cf. A008277.

Programs

Maple

A000770:=1/(z-1)/(6*z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[1/720 * (6^n - 6 * 5^n + 15 * 4^n - 20 * 3^n + 15 * 2^n - 6), {n, 6, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)
StirlingS2[Range[6, 25], 6] (* Alonso del Arte, Dec 07 2014 *)

Formula

G.f.: x^6/product(1 - k*x, k = 1..6).

E.g.f.: ((exp(x) - 1)^6)/6!.

a(n) = 1/720*(6^n - 6*5^n + 15*4^n - 20*3^n + 15*2^n - 6). - Vaclav Kotesovec, Nov 19 2012

a(n) = det(|s(i+6,j+5)|, 1 <= i,j <= n-6), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013