A002674 a(n) = (2n)!/2. (Formerly M4879 N2092)

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

Offset

1,2

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

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.

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)

Example

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

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 *)

Prog

(Magma) [n*Factorial(2*n-1): n in [1..15]]; // Vincenzo Librandi, Aug 23 2013
(PARI) a(n) = (2*n)!/2; \\ Indranil Ghosh, Apr 18 2017

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.

Sequence in context: A012631 A327986 A279182 * A012552 A012385 A012430

Adjacent sequences: A002671 A002672 A002673 * A002675 A002676 A002677

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved