[seqfan] Re: A033791

Neil Sloane njasloane at gmail.com
Mon Oct 20 03:38:26 CEST 2014


There were some questions on this list about why A033791 is important.

It is defined as (essentially) Product_{ d divides 32 } theta_2(q^d).

It is part of a series of sequences of the form
Product_{d | M } theta_i(q^d)
for various values of M, and i = 2, 3 and 4.

See for example A033761-A033807 and adjacent clumps of sequences.

I considered these sequences were worth putting into the OEIS because of
the many identities involving products of
the Dedekind eta functions - described in Fine's book on Basic
Hypergeometric Series, and in many other classical books.

I thought it was important to have all these products in the OEIS.  The
early ones in the series have come up in several places, and the others
might too.  These series arise in number theory, modular form theory, etc.

Do not think of A033761 as "obsolete". It is part of a large
family which are in the OEIS for a good reason.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Thu, Oct 16, 2014 at 7:31 PM, Wouter Meeussen <wouter.meeussen at telenet.be
> wrote:

> "    ...  while A033791 has a generating function Product(f(q^2^n)), where
> n goes from 0 to 5. "
>
> This choice of "5" is an arbitrary truncation of  an intrinsically
> infinite product
> t(z^2) . t(z^2^2) . t(z^2^3) . etc
> The OEIS has a rule that arbitrary constants should be avoided, especially
> so if more general definition is available.
>
> But, historically, I suppose the truncation to "5" was for calculation
> purposes  : q^2^5 = q^32 gives the sequence terms of A033791  up to  64
> terms.  Or, possibly its limitation to divisors of 32 had some
> lattice-theory foundation?
>
> My advice is to submit a new sequence where the limitation to 5 factors is
> replaced,
> and link the partitions comment to that one. Cross-referencing should then
> point out back to the
> older (and by then obsolete?) A033791.
>
> t(z^2) == Sum[(z^2)^(i (i+1)/2),{i, 0, inf]}]  ==   Sum[ z^(i (i + 1)),
> {i, 0, inf]}]  ==  EllipticTheta[2, 0, z] / (2*z^(1/4))
>
> Wouter.
>
> -----Original Message----- From: David Newman
> Sent: Friday, October 17, 2014 12:13 AM
>
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A033791
>
> Thank you Wouter.
>
> In answer to your question:
>
> "if I interpret your definition of carry-free frequencies correctly, then
> the number of carry-free partitions of n equals A018819
> "Binary partition function: number of partitions of n into powers of 2".
> Can you confirm?"
>
> Yes, I confirm this.
>
> Here's how I see the situation:
> If I let f(q)= 1+q+q^3+q^6+..., then my proposed sequence has generating
> function  Product(f(q^2^n)), where n goes from 0 to infinity, while A033791
> has generating function Product(f(q^2^n)), where n goes from 0 to 5.  I'm
> still not sure if to propose this as a new sequence or as a comment to the
> existing sequence since the two are identical for 60+ terms.
>
>
>
> On Thu, Oct 16, 2014 at 8:59 AM, Wouter Meeussen <
> wouter.meeussen at telenet.be
>
>> wrote:
>>
>
>  David,
>>
>> if I interpret your definition of carry-free frequencies correctly, then
>> the number of carry-free partitions of n equals A018819
>> "Binary partition function: number of partitions of n into powers of 2".
>> Can you confirm?
>>
>> It is quite easy to extend both sequences to 128 terms, and they are
>> perfectly equal.
>> In the light of the above connection to A018819, the conjectured equality
>> seems plausible if
>> you consider the implementation of the theta_2 in terms of products of
>> x^2^d.
>>
>> In Mathematica v. 10.0.1 : (*honni soit qui mal y pense *)
>>
>> carryfree[{li__Integer}]:=Block[{m,freq}, freq=Sort[Map[Last,Tally[{li}]
>> ]];m=Max[freq];Max[Total[IntegerDigits[freq,2,1+Floor[Log[2,m]]]]]<2];
>> your definition :
>> Table[Count[IntegerPartitions[n,All, Table[ k (k+1) 2,{k,2+Floor[Sqrt[2n
>> ]]}]],q_/;carryfree[q]],{n,128}]
>> {1,1,2,1,2,3,3,2,3,4,3,6,5,5,7,6,6,7,6,9,10, ...
>> ,237,221,220,246,225,242,262,243,249}
>>
>> same as A033791 :
>>
>> t[z_]:=Sum[z^(i (i+1)/2),{i,0,16}];
>> CoefficientList[Series[Product[t[x^2^d],{d,0,10}],{x,0,128}],x]
>>
>> you just need to properly adapt the limits in the definition of t[z] (upto
>> 16) and in the product t[x^2^d] (up to 10) over d .
>>
>> Nice find! Keep up the good work, David!
>>
>> Wouter.
>>
>>
>> -----Original Message----- From: David Newman
>> Sent: Thursday, October 16, 2014 3:16 AM
>> To: Sequence Fanatics Discussion list
>> Subject: [seqfan] Re: A033791
>>
>> The sequence that I'm lookinh at is the number of partitions into summands
>> which are triangular numbers and frequencies satisfying the "no binary
>> carry" condition. A partition has the "no binary carry" condition if the
>> sum of all its frequencies, when written in binary notation has no carry.
>>
>> On Wed, Oct 15, 2014 at 5:58 PM, Frank Adams-Watters <
>> franktaw at netscape.net>
>> wrote:
>>
>>  It would be easier to answer this question if you told us what your
>>
>>> proposed sequence is.
>>>
>>> Franklin T. Adams-Watters
>>>
>>>
>>> -----Original Message-----
>>> From: David Newman <davidsnewman at gmail.com>
>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>> Sent: Wed, Oct 15, 2014 4:52 pm
>>> Subject: [seqfan] Re: A033791
>>>
>>>
>>> The sequence that I'm working with will agree with A033791 for n<64, but
>>> will be different thereafter.  Should I propose it as new sequence or as
>>> a
>>> comment on A033791.
>>>
>>> On Wed, Oct 15, 2014 at 12:32 PM, Andrew N W Hone <A.N.W.Hone at kent.ac.uk
>>> >
>>> wrote:
>>>
>>>  Hi David,
>>>
>>>
>>>> I am not looking at the sequence just now but I guess theta_2 is a
>>>>
>>>>  theta
>>>>
>>>
>>>  function in Jacobi's classical notation: see
>>>
>>>>
>>>> http://en.wikipedia.org/wiki/Theta_function
>>>>
>>>> under "auxiliary functions". In the combinatorial setting, counting
>>>> partitions, you want to set the first argument z=0, which gives the
>>>>
>>>>  q-series
>>>>
>>>
>>>
>>>  \sum q^{(n+1/2)^2}
>>>>
>>>> where the sum is from n=-infinity to +infinity.
>>>>
>>>> A good classical reference is A Course of Modern Analysis by
>>>>
>>>>  Whittaker &
>>>>
>>>
>>>  Watson. The first volume of Mumford's Tata Lectures on Theta is also
>>>
>>>>
>>>>  great,
>>>>
>>>
>>>  and has a more modern point of view.
>>>
>>>>
>>>> All the best,
>>>> Andy
>>>>
>>>> ________________________________________
>>>> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of David
>>>>
>>>>  Newman [
>>>>
>>>
>>>  davidsnewman at gmail.com]
>>>
>>>> Sent: 15 October 2014 17:23
>>>> To: Sequence Fanatics Discussion list
>>>> Subject: [seqfan] A033791
>>>>
>>>> I have a partition type sequence which matches A033791 for the first
>>>>
>>>>  40
>>>>
>>>
>>>  terms.  However, beyond the computational evidence I have no reason to
>>>
>>>> think that the two are the same.  In fact, I'm guessing that they
>>>>
>>>>  differ
>>>>
>>>
>>>  for n>63.  I am hampered in my efforts  in part because I don't know
>>>
>>>>
>>>>  which
>>>>
>>>
>>>  function is meant by theta2.  Could someone give me the definition of
>>>
>>>> theta2 or point me to a source for the definition?
>>>>
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>>
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