[seqfan] checking a sequence before submission

David Newman davidsnewman at gmail.com
Mon Dec 13 16:37:25 CET 2010


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.

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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