A059332 Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.

1, 1, 2, 24, 3456, 9953280, 859963392000, 3120635156889600000, 634153008009974906880000000, 9278496603801318870491332608000000000, 12218100099725239100847669366019325952000000000000, 1769792823810713244721831122736499011207487815680000000000000000

Offset: 0

Author

Noam Katz (noamkj(AT)hotmail.com), Jan 26 2001

Keywords

• nonn

Comments

Hankel transform of n! (A000142(n)) and of A003319. - Paul Barry, Oct 07 2008

Hankel transform of A000255. - Paul Barry, Apr 22 2009

Monotonic magmas of size n, i.e., magmas with elements labeled 1..n where product(i,j) >= max(i,j). - Chad Brewbaker, Nov 03 2013

Also called the bouncing factorial function. - Alexander Goebel, Apr 08 2020

Examples

			a(4) = 3456 because the relevant matrix is {1 2 6 24 / 2 6 24 120 / 6 24 120 720 / 24 120 720 5040 } and the determinant is 3456.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..32
• Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
• Googology Wiki, Bouncing Factorial

Crossrefs

Cf. A010790, A385867.

Cf. A162014 and A055209. - Johannes W. Meijer, Jun 27 2009

Programs

Maple

with(linalg): Digits := 500: A059332 := proc(n) local A, i, j: A := array(1..n,1..n): for i from 1 to n do for j from 1 to n do A[i,j] := (i+j-1)! od: od: RETURN(det(A)) end: for n from 1 to 20 do printf(`%d,`, A059332(n)) od;
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)*n!^2/n)
    end:
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 29 2020

Mathematica

Table[n! BarnesG[n+1]^2, {n, 1, 10}] (* Jean-François Alcover, Sep 19 2016 *)

PARI

A059332(n)=matdet(matrix(n,n,i,j,(i+j-1)!)) \\ M. F. Hasler, Nov 03 2013

PARI

a(n) = 2^binomial(n,2)*prod(k=1,n-1, binomial(k+2,2)^(n-1-k)) \\ Ralf Stephan, Nov 04 2013

Ruby

def mono_choices(a,b,n)
    n - [a,b].max
end
def all_mono_choices(n)
    accum =1
    0.upto(n-1) do |i|
        0.upto(n-1) do |j|
            accum = accum * mono_choices(i,j,n)
        end
    end
    accum
end
1.upto(12) do |k|
puts all_mono_choices(k)
end # Chad Brewbaker, Nov 03 2013

Formula

a(n) = a(n-1)*(n!)*(n-1)! for n >= 2 so a(n) = product k=1, 2, ..., n k!*(k-1)!.

a(n) = 2^C(n,2)*Product_{k=1..(n-1), C(k+2,2)^(n-1-k)}. - Paul Barry, Jan 15 2009

a(n) = n!*product(k!, k=0..n-1)^2. - Johannes W. Meijer, Jun 27 2009

a(n) ~ (2*Pi)^(n+1/2) * exp(1/6 - n - 3*n^2/2) * n^(n^2 + n + 1/3) / A^2, where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 01 2015

Extensions

More terms from James Sellers, Jan 29 2001

Offset corrected. Comment and formula aligned with new offset by Johannes W. Meijer, Jun 24 2009

a(0)=1 prepended by Alois P. Heinz, Apr 08 2020