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A002674
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a(n) = (2n)!/2.
(Formerly M4879 N2092)
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29
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1, 12, 360, 20160, 1814400, 239500800, 43589145600, 10461394944000, 3201186852864000, 1216451004088320000, 562000363888803840000, 310224200866619719680000, 201645730563302817792000000, 152444172305856930250752000000, 132626429906095529318154240000000
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OFFSET
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1,2
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COMMENTS
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Right side of the binomial sum n-> sum( (-1)^i * (n-i)^(2*n) * binomial(2*n, i), i=0..n). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
a(n) is the number of ways to display n distinct flags on n distinct poles and then linearly order all (including any empty) poles. - Geoffrey Critzer, Dec 16 2009
Product of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 03 2013
Let f(x) be a polynomial in x. The expansion (2*sinh(x/2))^2 = x^2 + (1/12)*x^4 + (1/360)*x^6 + ... leads to the second central difference formula f(x+1) - 2*f(x) + f(x-1) = (2*sinh(D/2))^2(f(x)) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + ..., where D denotes the differential operator d/dx. - Peter Bala, Oct 03 2019
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REFERENCES
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A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.33)
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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FORMULA
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E.g.f.: (hypergeom([1/2, 1], [], 4*x)-1)/2 (cf. A090438).
Series reversion ( Sum_{n >= 1} x^n/a(n) ) = Sum_{n >= 1} (-1)^n*x^n/b(n-1), where b(n) = A002544(n). - Peter Bala, Apr 18 2017
Sum_{n>=1} 1/a(n) = 2*(cosh(1) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - cos(1)). (End)
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EXAMPLE
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a(3) = 360, since 2(3) = 6 has exactly 3 partitions into two parts: (5,1), (4,2), (3,3). Multiplying all the parts in the partitions, we get 5! * 3 = 360. - Wesley Ivan Hurt, Jun 03 2013
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MAPLE
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MATHEMATICA
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PROG
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CROSSREFS
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a(n) = A090438(n, 2), n >= 1 (first column of (4, 2)-Stirling2 array).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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