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A000481 Stirling numbers of the second kind, S(n,5).
(Formerly M4981 N2141)
7
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; text; internal format)
OFFSET

5,2

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=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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565

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 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]

MAPLE

A000481:=-1/(z-1)/(4*z-1)/(-1+3*z)/(2*z-1)/(5*z-1); [Conjectured by Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

lst={}; Do[f=StirlingS2[n, 5]; AppendTo[lst, f], {n, 5, 5!}]; lst [From 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 *)

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 Vladimir Kruchinin, Aug 30 2010]

CROSSREFS

a(n)= A008277(n, 5) (Stirling2 triangle).

Cf. A008277.

Sequence in context: A035330 A002803 A056281 * A215765 A055903 A026859

Adjacent sequences:  A000478 A000479 A000480 * A000482 A000483 A000484

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Sean A. Irvine, Nov 14 2010

STATUS

approved

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Last modified April 24 03:19 EDT 2014. Contains 240947 sequences.