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

1, 12, 360, 20160, 1814400, 239500800, 43589145600, 10461394944000, 3201186852864000, 1216451004088320000, 562000363888803840000, 310224200866619719680000, 201645730563302817792000000, 152444172305856930250752000000, 132626429906095529318154240000000

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

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

Examples

			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
		

References

• 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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• Ronald P. Nordgren, Compound Lucas Magic Squares, arXiv:2103.04774 [math.GM], 2021. See Table 2 p. 12.
• 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. E. Salzer, Annotated scanned copy of left side of Table II.
• Eric Weisstein's World of Mathematics, Central Difference.

Crossrefs

a(n) = A090438(n, 2), n >= 1 (first column of (4, 2)-Stirling2 array).

Cf. A000142, A010050, A090438, A002544. Cf. A002671, A002672, A002673, A002675, A002676, A002677.

Programs

Magma

[n*Factorial(2*n-1): n in [1..15]]; // Vincenzo Librandi, Aug 23 2013

Maple

seq((2*k)!/2, k=1..20); # Wesley Ivan Hurt, Aug 22 2013

Mathematica

Table[n! Pochhammer[n, n], {n, 0, 10}] (* Geoffrey Critzer, Dec 16 2009 *)
Table[(2 n)! / 2, {n, 1, 15}] (* Vincenzo Librandi, Aug 23 2013 *)

PARI

a(n) = (2*n)!/2; \\ Indranil Ghosh, Apr 18 2017

Formula

4*sinh(x/2)^2 = Sum_{k>=1} x^(2k)/a(k). - Benoit Cloitre, Dec 08 2002

E.g.f.: (hypergeom([1/2, 1], [], 4*x)-1)/2 (cf. A090438).

a(n) = n*(2n-1)!. - Geoffrey Critzer, Dec 16 2009

a(n) = A010050(n)/2. - Wesley Ivan Hurt, Aug 22 2013

a(n) = Product_{k=0..n-1} (n^2 - k^2). - Stanislav Sykora, Jul 14 2014

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

From Amiram Eldar, Jul 09 2020: (Start)

Sum_{n>=1} 1/a(n) = 2*(cosh(1) - 1).

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - cos(1)). (End)