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%I M4981 N2141 #70 Feb 13 2023 07:58:37
%S 1,15,140,1050,6951,42525,246730,1379400,7508501,40075035,210766920,
%T 1096190550,5652751651,28958095545,147589284710,749206090500,
%U 3791262568401,19137821912055,96416888184100,485000783495250,2436684974110751,12230196160292565,61338207158409090
%N Stirling numbers of the second kind, S(n,5).
%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="/A000481/b000481.txt">Table of n, a(n) for n=5..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=348">Encyclopedia of Combinatorial Structures 348</a>
%H Vladimir Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%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 a(n) = A008277(n, 5) (Stirling2 triangle).
%F G.f.: x^5/product(1-k*x, k=1..5).
%F E.g.f.: ((exp(x)-1)^5)/5!.
%F 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
%F 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
%p A000481:=-1/(z-1)/(4*z-1)/(-1+3*z)/(2*z-1)/(5*z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p a := n -> (1-4^n+2*(3^n-2^n)+5^(n-1))/24:
%p seq(a(n), n=5..29); # _Peter Luschny_, May 09 2015
%t lst={};Do[f=StirlingS2[n, 5];AppendTo[lst, f], {n, 5, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)
%t 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 *)
%t StirlingS2[Range[5,30],5] (* _Harvey P. Dale_, May 15 2017 *)
%Y Cf. A008277.
%K nonn
%O 5,2
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Nov 14 2010
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