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%I #56 Apr 29 2020 07:52:45
%S 1,1,2,24,3456,9953280,859963392000,3120635156889600000,
%T 634153008009974906880000000,9278496603801318870491332608000000000,
%U 12218100099725239100847669366019325952000000000000,1769792823810713244721831122736499011207487815680000000000000000
%N Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.
%C Hankel transform of n! (A000142(n)) and of A003319. - _Paul Barry_, Oct 07 2008
%C Hankel transform of A000255. - _Paul Barry_, Apr 22 2009
%C 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
%C Also called the bouncing factorial function. - _Alexander Goebel_, Apr 08 2020
%H Alois P. Heinz, <a href="/A059332/b059332.txt">Table of n, a(n) for n = 0..32</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H Googology Wiki, <a href="https://googology.wikia.org/wiki/Bouncing_Factorial">Bouncing Factorial</a>
%F a(n) = a(n-1)*(n!)*(n-1)! for n >= 2 so a(n) = product k=1, 2, ..., n k!*(k-1)!.
%F a(n) = 2^C(n,2)*Product_{k=1..(n-1), C(k+2,2)^(n-1-k)}. - _Paul Barry_, Jan 15 2009
%F a(n) = n!*product(k!, k=0..n-1)^2. - _Johannes W. Meijer_, Jun 27 2009
%F 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
%e 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.
%p 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;
%p # second Maple program:
%p a:= proc(n) option remember;
%p `if`(n=0, 1, a(n-1)*n!^2/n)
%p end:
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Apr 29 2020
%t Table[n! BarnesG[n+1]^2, {n, 1, 10}] (* _Jean-François Alcover_, Sep 19 2016 *)
%o (Ruby)
%o def mono_choices(a,b,n)
%o n - [a,b].max
%o end
%o def all_mono_choices(n)
%o accum =1
%o 0.upto(n-1) do |i|
%o 0.upto(n-1) do |j|
%o accum = accum * mono_choices(i,j,n)
%o end
%o end
%o accum
%o end
%o 1.upto(12) do |k|
%o puts all_mono_choices(k)
%o end # _Chad Brewbaker_, Nov 03 2013
%o (PARI) A059332(n)=matdet(matrix(n,n,i,j,(i+j-1)!)) \\ _M. F. Hasler_, Nov 03 2013
%o (PARI) a(n) = 2^binomial(n,2)*prod(k=1,n-1, binomial(k+2,2)^(n-1-k)) \\ _Ralf Stephan_, Nov 04 2013
%Y Cf. A010790.
%Y Cf. A162014 and A055209. - _Johannes W. Meijer_, Jun 27 2009
%K nonn
%O 0,3
%A Noam Katz (noamkj(AT)hotmail.com), Jan 26 2001
%E More terms from _James A. Sellers_, Jan 29 2001
%E Offset corrected. Comment and formula aligned with new offset by _Johannes W. Meijer_, Jun 24 2009
%E a(0)=1 prepended by _Alois P. Heinz_, Apr 08 2020
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