A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

1, 1, 12, 8640, 870912000, 22122558259200000, 222531556847250309120000000, 1280394777025250130271722799104000000000, 5746332926632566442385615219551212618645504000000000000

Offset: 0

Author

Henry Bottomley, May 14 2005

Keywords

• nonn

Comments

Inverse product of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 12 2006

The n X n matrix with A(i,j) = 1/(i+j-1)! (i,j = 1..n) has determinant (-1)^floor(n/2)/a(n). - Mikhail Lavrov, Nov 01 2022

Examples

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*1!*2!) = 34560/4 = 8640.
n = 2: HilbertMatrix[n,n]
  1/1 1/2
  1/2 1/3
so a(2) = 1 / (1 * 1/2 * 1/2 * 1/3) = 12.
The n X n Hilbert matrix begins:
  1/1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
  1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20
• Mathematics Stack Exchange, Determinant of a matrix involving factorials.
• Eric Weisstein's World of Mathematics, Hilbert Matrix.

Crossrefs

Cf. A000178, A002457, A005249, A098118.

Programs

Magma

A107254:= func< n | n eq 0 select 1 else (&*[Factorial(n+j)/Factorial(j): j in [0..n-1]]) >;
[A107254(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

Maple

a:= n-> mul((n+i)!/i!, i=0..n-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 23 2012

Mathematica

Table[Product[(i+j-1),{i,1,n},{j,1,n}], {n,1,10}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[n!*BarnesG[2n+1]/(BarnesG[n+2]*BarnesG[n+1]), {n,0,12}] (* G. C. Greubel, Apr 21 2021 *)

Sage

a = lambda n: prod(rising_factorial(k,n) for k in (1..n))
print([a(n) for n in (0..10)]) # Peter Luschny, Nov 29 2015

Formula

a(n) = n!*(n+1)!*(n+2)!*...*(2n-1)!/(0!*1!*2!*3!*...*(n-1)!) = A000178(2n-1)/A000178(n-1)^2 = A079478(n)/A000984(n) = A079478(n-1)*A009445(n-1) = A107252(n)*A000142(n) = A088020(n)/A039622(n).

a(n) = 1/Product_{j=1..n} ( Product_{i=1..n} 1/(i+j-1) ). - Alexander Adamchuk, Apr 12 2006

a(n) = 2^(n*(n-1)) * A136411(n) for n > 0 . - Robert Coquereaux, Apr 06 2013

a(n) = A136411(n) * A053763(n) for n > 0. [Following remark from Robert Coquereaux] - M. F. Hasler, Apr 06 2013

a(n) ~ A * 2^(2*n^2-1/12) * n^(n^2+1/12) / exp(3*n^2/2+1/12), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015

a(n) = Product_{k=1..n} rf(k,n) where rf denotes the rising factorial. - Peter Luschny, Nov 29 2015

a(n) = (n! * G(2*n+1))/(G(n+1)*G(n+2)), where G(n) is the Barnes G - function. - G. C. Greubel, Apr 21 2021