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A000481
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Stirling numbers of the second kind, S(n,5).
(Formerly M4981 N2141)
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14
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1, 15, 140, 1050, 6951, 42525, 246730, 1379400, 7508501, 40075035, 210766920, 1096190550, 5652751651, 28958095545, 147589284710, 749206090500, 3791262568401, 19137821912055, 96416888184100, 485000783495250, 2436684974110751, 12230196160292565, 61338207158409090
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OFFSET
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5,2
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REFERENCES
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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).
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LINKS
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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
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A008277.
Sequence in context: A346977 A354398 A056281 * A327506 A346955 A346920
Adjacent sequences: A000478 A000479 A000480 * A000482 A000483 A000484
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Sean A. Irvine, Nov 14 2010
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STATUS
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approved
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