cp's OEIS Frontend

A002674 a(n) = (2n)!/2.

%I A002674 M4879 N2092 #64 Feb 16 2025 08:32:26
%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, <a href="/A002674/b002674.txt">Table of n, a(n) for n = 1..100</a>
%H A002674 Ronald P. Nordgren, <a href="https://arxiv.org/abs/2103.04774">Compound Lucas Magic Squares</a>, arXiv:2103.04774 [math.GM], 2021. See Table 2 p. 12.
%H A002674 H. E. Salzer, <a href="https://doi.org/10.1002/sapm1963421162">Tables of coefficients for obtaining central differences from the derivatives</a>, 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, <a href="/A002675/a002675.png">Annotated scanned copy of left side of Table II</a>.
%H A002674 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CentralDifference.html">Central Difference</a>.
%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_