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
KEYWORD
nonn,easy,changed
AUTHOR
STATUS
approved