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%I M5112 N2215 #51 Apr 13 2022 13:25:15
%S 1,21,266,2646,22827,179487,1323652,9321312,63436373,420693273,
%T 2734926558,17505749898,110687251039,693081601779,4306078895384,
%U 26585679462804,163305339345225,998969857983405,6090236036084530,37026417000002430,224595186974125331
%N Stirling numbers of the second kind, S(n,6).
%D 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.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%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).
%H T. D. Noe, <a href="/A000770/b000770.txt">Table of n, a(n) for n=6..200</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=349">Encyclopedia of Combinatorial Structures 349</a>
%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
%F G.f.: x^6/product(1 - k*x, k = 1..6).
%F E.g.f.: ((exp(x) - 1)^6)/6!.
%F a(n) = 1/720*(6^n - 6*5^n + 15*4^n - 20*3^n + 15*2^n - 6). - _Vaclav Kotesovec_, Nov 19 2012
%F 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
%p 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
%t 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 *)
%t StirlingS2[Range[6, 25], 6] (* _Alonso del Arte_, Dec 07 2014 *)
%Y a(n)= A008277(n, 6) (Stirling2 triangle).
%Y Cf. A008277.
%K nonn
%O 6,2
%A _N. J. A. Sloane_
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