Re: Recursion with incredible divisibillity properties

On Mar 19, 9:29 am, vno...@xxxxxxxxx wrote:
Hi All,

define integers d(mu, k) recursively by:

d(mu, 0) = 1
d(mu, k) = (mu - k +1)d(mu + 1, k - 1) - (k -1)(mu + 1) d(mu + 2, k
-2).

It seems that:

1) For k greater/equal 1, d(mu, k) is divisible by every positive
integers less than or equal to k.
2) For k odd, d(m, k) is divisible by mu(m+2)(mu+4)...(mu + k - 1)

Has anyone ever seen a similar set of numbers? Can anyone help me
prove this? I have an 'abstract nonsense proof' of the second claim in
case mu is even, using the representation theory of SL(2, R), but I
would really like to see an 'ordinary' proof using numbers...

Any ideas?

Thanks!

d[m,k] is a polynomial of degree k in m. E.g., d[m,2] = m^2 - m - 2.
Here are the coefficients (found by Mathematica) for k = 0...9:

0: 1
1: 0 1
2: -2 -1 1
3: 0 -10 -3 1
4: 36 18 -25 -6 1
5: 0 344 130 -45 -10 1
6: -1800 -840 1504 495 -65 -15 1
7: 0 -25488 -10164 4144 1365 -77 -21 1
8: 176400 77280 -164060 -60172 7889 3080 -70 -28 1
9: 0 3302784 1323792 -635860 -241164 9345 6048 -30 -36 1

.