OFFSET
0,3
COMMENTS
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
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
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
EXAMPLE
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.
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Noam Katz (noamkj(AT)hotmail.com), Jan 26 2001
EXTENSIONS
More terms from James A. 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
STATUS
approved