A067700

[Robert G. Wilson v]

Joseph and others,

	Here is what I wrote 7 Feb 02:

"Neil,

         Do the leading digits belong here? Also the formula line %F
A067700 a(n) = 2*(n^2)!* Product_{k = 0 .. n-1 } k!/(n+k)!. does not
produce the sequence. I wrote a[n_] := 2*(n^2)!*Product[k!/(n + k)!, {k,
1, n - 1}]; Table[a[n], {n, 1, 10}] and got:

2, 8, 504, 1153152, 168275764800, 2407165968578342400,
4788742737385049982623884800, 1780642079411485280163076498360356864000,
159943989198524502594920793284078996733117111490560000,
4353607386405822605116660595502838129080647848043621660449907712000000,

which is no where close.

         If on the other hand the formula was a(n) = 2*(n^2)!/n!*
Product_{k = 0 .. n-1 } k!/(n+k)!, the coding is a[n_] :=
2*(n2)!/n!*Product[k!/(n + k)!, {k, 1, n - 1}]; Table[a[n], {n, 1, 10}]
. Then we get starting with an offset of one:

2, 4, 84, 48048, 1402298040, 3343286067469920, 950147368528779758457120,
44162749985403900797695349661715200,
440762756830149092247907829817237094171949712000,
1199737485230881449822712906609027262202559481934419549286240000,

         That at least explains the leading zeroes. Bob."

	I can not say what the original sequence was nor its length.

Thanks, Bob.



Joseph S. Myers wrote:
> In sequence A067700, the formula given as the name of the sequence, the
> numbers given for the sequence and the description given in the comments
> yield three different sequences.
> 
> The name is "a(n) = 2*(n^2)!/n!*Product_{k = 0 .. n-1 } k!/(n+k)!.".  This
> yields:
> 
> 2,2,14,2002,11685817,4643452871486,188521303279519793344,
> 1095306299241168174546015616610,1214623999201248600771350941956671886496775
> 
> The sequence given however is:
> 2,4,84,48048,1402298040,3343286067469920,950147368528779758457120,
> 44162749985403900797695349661715200,
> 440762756830149092247907829817237094171949712000
> 
> (which looks like it matches the formula if "/n!" is removed).
> 
> The comments say "Number of distinct-integer n^2-arrangements(or
> n^2-tuples) free of any monotonic increasing or decreasing
> (n+1)-subsequence.".  This sequence, however, starts 1,4,1764,577152576.
> This looks like it may match the formula if the "2*" and "/n!" are both
> removed, then the result is squared (yielding "a(n) =
> ((n^2)!*Product_{k = 0 .. n-1 } k!/(n+k)!)^2."), though I don't have a
> proof of this.
>