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A059332
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Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.
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8
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1, 1, 2, 24, 3456, 9953280, 859963392000, 3120635156889600000, 634153008009974906880000000, 9278496603801318870491332608000000000, 12218100099725239100847669366019325952000000000000, 1769792823810713244721831122736499011207487815680000000000000000
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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) ~ (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
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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PROG
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(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)
(PARI) a(n) = 2^binomial(n, 2)*prod(k=1, n-1, binomial(k+2, 2)^(n-1-k)) \\ Ralf Stephan, Nov 04 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Noam Katz (noamkj(AT)hotmail.com), Jan 26 2001
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EXTENSIONS
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Offset corrected. Comment and formula aligned with new offset by Johannes W. Meijer, Jun 24 2009
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STATUS
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approved
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