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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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7
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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, 307440364830580800, 1540200411172850701, 7713000216608565075, 38613005164147863680, 193257076459811283150, 967053687799836580251, 4838341969836854217585, 24204004899040755811870, 121069207474151411298300, 605542777625121255411401, 3028500874158801262498095, 15145652389974035183277660, 75740854251732106906082250, 378754641141533842447776151, 1893974687265439470662014605
(list; graph; refs; listen; history; internal format)
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OFFSET
| 5,2
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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).
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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
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565
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FORMULA
| 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. [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Aug 30 2010]
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MAPLE
| A000481:=-1/(z-1)/(4*z-1)/(-1+3*z)/(2*z-1)/(5*z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| lst={}; Do[f=StirlingS2[n, 5]; AppendTo[lst, f], {n, 5, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 27 2008]
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x)), {x, 0, 25}], x] (* From Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
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PROG
| (Other) 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); (for Maxima) [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Aug 30 2010]
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CROSSREFS
| a(n)= A008277(n, 5) (Stirling2 triangle).
Cf. A008277.
Sequence in context: A035330 A002803 A056281 * A055903 A026859 A096046
Adjacent sequences: A000478 A000479 A000480 * A000482 A000483 A000484
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Sean A. Irvine (sairvin(AT)xtra.co.nz), Nov 14 2010
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