[seqfan] Re: checking a sequence before submission

[Joerg Arndt]

* David Newman <davidsnewman at gmail.com> [Dec 13. 2010 17:13]:
> Hi Sequence Fans,
> 
>      I'd like someone to check a sequence for me before I submit it,
> having had the unpleasant experience of submitting an incorrect
> sequence in the past.
> 
>      For a given positive integer, n, let S_n be the set of partitions
> of n into distinct parts where the number of parts is maximal for that
> n.  For example, for n=6, the set S_6 consists of just one such
> partition S_6={1,2,3}.  Similarly, for n=7, S_7={1,2,4}, But for n=8,
> S_8 will contain two partitions S_8= { {1,2,5}, {1,3,4} }.
> 
>      Now form the sum 1+ x/(1-x) + x^2/(1-x^2) + x^3/ ( ( 1-x)
> (1-x^2)) + x^4/ ( ( 1-x) (1-x^3) ) +  x^5/  ( (1-x) (1-x^4) ) + x^5 (
> ( 1-x^2) (1-x^3)) + x^6/ ( ( 1-x) (1-x^2) (1-x^3)) + ...
> 
> whose general term is x^n divided by the product
> (1-x^(p_1))...(1-x^(p_i))  where  the p's  come from the partitions in
> S_n.

How would the term for n=8 be defined?

All partitions of n=8 into distinct parts are:
   1:  1+2+5  ***
   2:  1+3+4  ***
   3:  1+7
   4:  2+6
   5:  3+5
   6:  8

Now should I
a) use all of 1,2,5, and 1,3,4  ?
b) use multiplicities, i.e. here (1-x^1)^2  ?

> 
> The sequence is the sequence of coefficients of this sum.
> 
> The numbers that I've gotten are
> 1,1,2,2,4,6,7,10,14,20,24,32,40,54,69,86,106,135,165,206,256,311,378,460,555,670,808,970,1156,1380,1638,1938,2296,2706,3188,3752,4390,5136
> 
> If anyone is willing to do the calculations I can make the Mathematica
> program which I used available to them .
> 
> 
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