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%I A002674 M4879 N2092 #63 Sep 08 2022 08:44:31
%S A002674 1,12,360,20160,1814400,239500800,43589145600,10461394944000,
%T A002674 3201186852864000,1216451004088320000,562000363888803840000,
%U A002674 310224200866619719680000,201645730563302817792000000,152444172305856930250752000000,132626429906095529318154240000000
%N A002674 a(n) = (2n)!/2.
%C A002674 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
%C A002674 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
%C A002674 Product of the partition parts of 2n into exactly two parts. - _Wesley Ivan Hurt_, Jun 03 2013
%C A002674 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
%D A002674 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)
%D A002674 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002674 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002674 Vincenzo Librandi, Table of n, a(n) for n = 1..100
%H A002674 Ronald P. Nordgren, Compound Lucas Magic Squares, arXiv:2103.04774 [math.GM], 2021. See Table 2 p. 12.
%H A002674 H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.
%H A002674 H. E. Salzer, Annotated scanned copy of left side of Table II.
%H A002674 Eric Weisstein's World of Mathematics, Central Difference.
%F A002674 4*sinh(x/2)^2 = Sum_{k>=1} x^(2k)/a(k). - _Benoit Cloitre_, Dec 08 2002
%F A002674 E.g.f.: (hypergeom([1/2, 1], [], 4*x)-1)/2 (cf. A090438).
%F A002674 a(n) = n*(2n-1)!. - _Geoffrey Critzer_, Dec 16 2009
%F A002674 a(n) = A010050(n)/2. - _Wesley Ivan Hurt_, Aug 22 2013
%F A002674 a(n) = Product_{k=0..n-1} (n^2 - k^2). - _Stanislav Sykora_, Jul 14 2014
%F A002674 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
%F A002674 From _Amiram Eldar_, Jul 09 2020: (Start)
%F A002674 Sum_{n>=1} 1/a(n) = 2*(cosh(1) - 1).
%F A002674 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - cos(1)). (End)
%e A002674 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
%p A002674 seq((2*k)!/2, k=1..20); # _Wesley Ivan Hurt_, Aug 22 2013
%t A002674 Table[n! Pochhammer[n, n], {n, 0, 10}] (* _Geoffrey Critzer_, Dec 16 2009 *)
%t A002674 Table[(2 n)! / 2, {n, 1, 15}] (* _Vincenzo Librandi_, Aug 23 2013 *)
%o A002674 (Magma) [n*Factorial(2*n-1): n in [1..15]]; // _Vincenzo Librandi_, Aug 23 2013
%o A002674 (PARI) a(n) = (2*n)!/2; \\ _Indranil Ghosh_, Apr 18 2017
%Y A002674 a(n) = A090438(n, 2), n >= 1 (first column of (4, 2)-Stirling2 array).
%Y A002674 Cf. A000142, A010050, A090438, A002544. Cf. A002671, A002672, A002673, A002675, A002676, A002677.
%K A002674 nonn,easy
%O A002674 1,2
%A A002674 _N. J. A. Sloane_, _Simon Plouffe_
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