# [seqfan] Re: A071816, A005900

franktaw at netscape.net franktaw at netscape.net
Thu Jul 23 22:03:28 CEST 2009

```I don't like the definition of A071816.  When I see "Number of 6-digit
numbers", I don't expect to see numbers less than 100000 counted; these
aren't 6 digit numbers, they are 5 (or fewer) digit numbers.

I would rather see the comment from Vladeta used as the definition
(although I would write " < n" instead of "<=n-1").  The current
definition would then become a comment, with the phrase "with leading
zeros allowed" appended.  This phrase should be appended to the current
definition, regardless.

-----
It is certainly correct that the sequence for order k will be a
polynomial of degree 2k-1.

-----Original Message-----
From: rhhardin at att.net

Generalizing
http://www.research.att.com/~njas/sequences/A005900  (order 2)
http://www.research.att.com/~njas/sequences/A071816  (order 3)

numerically out to order 50, the ones I could compute enough terms for
to check
turned
out to be odd polynomials in n of degree 2*order-1.

The defining problem I used, eg., for order 8, just to choose one, was

a(n) = Number of ways the component sums of two 1..n 8-vectors can be
equal

The polynomial for order 8 is empirically

a(n) = (2330931341/6810804000)*n^15 + (56057/340200)*n^13
+ (404711/3402000)*n^11 + (227197/2381400)*n^9 + (546533/6804000)*n^7
+ (130231/1871100)*n^5 + (266681/4299750)*n^3 + (1/15)*n

Transposing the resulting table a(order,n) there are two existing series

http://www.research.att.com/~njas/sequences/A082758 (n=3)
http://www.research.att.com/~njas/sequences/A005721 (n=4)

with the result that, eg., with the obvious generalization,

a(order,n) = largest coefficient of (1+...+x(n-1)) ^ (2*order)

and the curious question, is it obvious that this largest coefficient
is an odd
polynomial in n of degree 2*order-1.

I hope I've done variable name substitutions correctly above.

--
rhhardin at mindspring.com
rhhardin at att.net (either)

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