# [seqfan] Re: A033791

Wouter Meeussen wouter.meeussen at telenet.be
Thu Oct 16 14:59:50 CEST 2014

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];
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.
>
>
>
> -----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?
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

_______________________________________________

Seqfan Mailing list - http://list.seqfan.eu/