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A000497
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S2(j,2j+2) where S2(n,k) is a 2-associated Stirling number of the second kind.
(Formerly M5186 N2254)
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2
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1, 25, 490, 9450, 190575, 4099095, 94594500, 2343240900, 62199262125, 1764494857125, 53338158823950, 1712934942468750, 58274046742786875, 2094379201311271875, 79318164037837725000, 3157886388887074845000
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OFFSET
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1,2
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
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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Vincenzo Librandi, Table of n, a(n) for n = 1..100
H. W. Gould, Harris Kwong, Jocelyn Quaintance, On Certain Sums of Stirling Numbers with Binomial Coefficients, J. Integer Sequences, 18 (2015), #15.9.6.
M. Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., 56 (1934), p. 87-95.
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FORMULA
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G.f.: x*(4*x+1)*hypergeom([3, 7/2],[],2*x)+28*x^3*hypergeom([4, 9/2],[],2*x). - Mark van Hoeij, Apr 07 2013
a(n) = n*(n+1)*(2*n+1)*2^n*GAMMA(n+3/2)/(9*sqrt(Pi)). - Vaclav Kotesovec, Aug 07 2013
(2*n-1)*(n-1)*a(n) -(n+1)*(1+2*n)^2*a(n-1)=0. - R. J. Mathar, Jun 09 2018
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MAPLE
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gf := (u, t)->exp(u*(exp(t)-1-t)); S2a := j->simplify(subs(u=0, t=0, diff(gf(u, t), u$j, t$(2*j+2)))/j!); for i from 1 to 20 do S2a(i); od;
# Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000
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MATHEMATICA
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t[n_, k_] := Sum[ (-1)^i*Binomial[n, i]*Sum[ (-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!), {j, 0, k-i}], {i, 0, k}]; Table[ t[2n+2, n], {n, 1, 16} ](* Jean-François Alcover, Feb 24 2012 *)
Table[n*(n+1)*(2*n+1)*2^n*Gamma[n+3/2]/(9*Sqrt[Pi]), {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2013 *)
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CROSSREFS
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Cf. A008299, A000504.
Sequence in context: A059946 A357147 A118445 * A353116 A028341 A282689
Adjacent sequences: A000494 A000495 A000496 * A000498 A000499 A000500
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KEYWORD
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nonn,nice,easy
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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 Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 12 2000
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STATUS
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approved
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