A034140 - A034150: formula

[Frank Ruskey]

It might be noted that these sequences are all unimodal
and that (1+x)(1+x^2)(1+x^3)...(1+x^n) is the rank
generating function of the poset M(n), well loved
by algebraic combinatorists.  See, e.g.,

Robert A. Proctor,
Solution of two difficult combinatorial problems
with linear algebra, American Mathematical Monthly,
1989, 721-734.

The middle (and largest coefficients) are A025591.
(On the links section of the A025591 page, the links
should be attached to the J. Int. Sequences rather than
the paper titles.)

The number of times the largest coefficient occurs
is A070937.  A table containing the A034140 - A034150
numbers as rows is A053632 (to bad it can't be
viewed as a table!).

Cheers,
Frank



seqfan at ext.jussieu.frzak seidov wrote:
> OK, 
> Gordon, Neil, seqfans,
> 
> in that case,
> 
> i'd suggest to 
> 
> a) keep only 
> 1+n(n+1)/2 
> nonzero (not non-zero!)
> terms, 
> with, e.g. n=10,11,...
> 
> b) add fini, full keys
> and 
> 
> c) add Mmca:
> CoefficientList[Product[(1 + x^i), {i, n}], x],
> for relevant n's.
> 
> Right?
> Zak
> 
> 
> --- Gordon Royle <gordon at csse.uwa.edu.au> wrote:
> 
>>> Expansion of (1+x)(1+x^2)(1+x^3)...
>>>
>>>Is it OK?
>>>
>>
>>It is not exactly incorrect, but it is very
>>ambiguous.
>>
>>The expressions referred to all START with those
>>those three terms, but 
>>they finish at different places...
>>
>>34140: Expansion of (1+x)(1+x^2)...(1+x^10)
>>34141: Expansion of (1+x)(1+x^2)...(1+x^10)(1+x^11)
>>34142: Expansion of
>>(1+x)(1+x^2)...(1+x^10)(1+x^11)(1+x^12)
>>
>>and so on..
>>


-- 
----------------------
Frank Ruskey             e-mail: (last_name)(AT)cs(DOT)uvic(DOT)ca
Dept. of Computer Science      fax:    250-721-7292
University of Victoria         office: 250-721-7232
Victoria, B.C. V8W 3P6 CANADA  WWW: http://www.cs.uvic.ca/~(last_name)