I'm puzzled by A129947

[N. J. A. Sloane]

> We have 2*(k-2)! = 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760 (all
> in A129947),

... all except the k=3 term?

I.e., there's a 3 where we'd expect 2*(3-2)! = 2
(and there's a 1 where we'd expect 2*(2-2)! = 2)

... and the offset is given as 1, yet k! / T(k-1) would be undefined at k = 
1.

> So is A129947 just 1 U 3 U 2*(n-2)! but accidently missing 2*(12-2)! I
> wonder? Or am I misunderstanding the definition?

It does look as though the k=12 term is missing, and something went wrong at 
the beginning of the sequence, as well....   ?:-/

Similarly puzzled,

-- Jon

----- Original Message ----- 
From: "Jonathan Post" <jvospost3 at gmail.com>
To: "sequence fans" <seqfan at ext.jussieu.fr>; "Giovanni Teofilatto" 
<g.teofilatto at tiscalinet.it>
Sent: Saturday, June 09, 2007 2:35 PM
Subject: I'm puzzled by A129947


> I'm puzzled by A129947 a(n) = k!/T(k-1), where k is a positive integer.
>
> If T(n) means n-th triangular number = A000217(n) = n*(n+1)/2 then we 
> have:
>
> k!/T(k-1) = [1 * 2 * 3 * 4 * ... * (k-1) * k] / [(k-1)*k/2]
> = 2 * [1 * 2 * 3 * 4 * ... *(k-1) * k] / [(k-1)*k]
> = 2 * [1 * 2 * 3 * 4 * ... (k-2)] = 2*(k-2)!
>
> We have 2*(k-2)! = 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760 (all
> in A129947),
>
> then 7257600 (not in A129947)
>
> then 79833600, 958003200, ...
>
> So is A129947 just 1 U 3 U 2*(n-2)! but accidently missing 2*(12-2)! I
> wonder? Or am I misunderstanding the definition?
>
> One might also look at the array A[k,n] = {n! / j-th k-gonal number for 
> some j}.
>
>