[seqfan] Re: Generalized Pascal triangle needs a recurrence
Neil Sloane
njasloane at gmail.com
Tue May 7 22:04:26 CEST 2013
PS I was asked to explain the connection with the Pascal triangle.
Here is my answer:
If you had pursued the cross-references, you would have seen the
connection with Pascal's triangle. There is a sequence of triangles,
with a parameter l (ell), given in the first column:
basic triangle odd-numbered rows
0 Pascal A7318 A034870
1 A159041 A171692
2 added just now A225076
3 missing! A225398
4 A225415
I was asking about A159041
If the question had been easy I would not have posted it!
On Tue, May 7, 2013 at 3:52 PM, Max Alekseyev <maxale at gmail.com> wrote:
> I think this is not enough to call something a generalization of
> Pascal's triangle.
> Something would be called generalization of Pascal's triangle if the
> latter could be obtained as a partial case (e.g., a larger
> triangle/table containing P's triangle, or triangle of polynomials
> whose free terms form P's triangle etc.)
> Regards,
> Max
>
> On Tue, May 7, 2013 at 3:47 PM, Maximilian Hasler
> <maximilian.hasler at gmail.com> wrote:
>> the common points are :
>> * triangle
>> * T(n,0)=T(n,n)=1
>> * T(n,k)=T(n,n-k)
>>
>> but what is imo characteristic for P's triangle is the relation
>> T(n,k)+T(n,k+1)=T(n+1,k+1) which is not satisfied,
>> nor with any other constant coefficients
>> and I don't think that it is with n-dependent coefficients (both should be
>> the same because of symmetry, but no such choice would work for the 3rd row
>> AFAICS, since c*(1 -25) = -56 => c= 23/24
>> but
>> c*(-25 -25)=246 => c=-123/25
>>
>> and when coefficients depend on both n and k, then there is no more
>> "economy" w.r.t. explicitly specifying the elements.
>>
>> Maximilian
>>
>>
>> On Tue, May 7, 2013 at 3:18 PM, Max Alekseyev <maxale at gmail.com> wrote:
>>>
>>> I do not see how it generalizes Pascal's triangle -- could you please
>>> explain this relationship?
>>> Thanks,
>>> Max
>>>
>>> On Tue, May 7, 2013 at 2:43 PM, Neil Sloane <njasloane at gmail.com> wrote:
>>> > Dear SeqFans,
>>> > Roger Bagula has constructed some interesting triangles that
>>> > generalize Pascal's triangle (A007318), but his constructions are
>>> > fairly complicated. It would be nice to have a direct construction
>>> > like that for Pascal's triangle.
>>> > The first of them is in A159041.
>>> >
>>> > This triangle begins:
>>> > {1},
>>> > {1, 1},
>>> > {1, -10, 1},
>>> > {1, -25, -25, 1},
>>> > {1, -56, 246, -56, 1},
>>> > {1, -119, 1072, 1072, -119, 1},
>>> > {1, -246, 4047, -11572, 4047, -246, 1},
>>> > {1, -501, 14107, -74127, -74127, 14107, -501, 1},
>>> > {1, -1012, 46828, -408364, 901990, -408364, 46828, -1012, 1},
>>> > {1, -2035, 150602, -2052886, 7685228, 7685228, -2052886, 150602, -2035,
>>> > 1},
>>> > { 1, -4082, 474189, -9713496, 56604978, -105907308, 56604978,
>>> > -9713496, 474189, -4082, 1}
>>> >
>>> > Can anyone see a recurrence? I can't.
>>> > (The second column is easy)
>>> > There's a Mma program that will produce more terms if they are needed.
>>> > Neil
>>> >
>>> > --
>>> > Dear Friends, I have now retired from AT&T. New coordinates:
>>> >
>>> > Neil J. A. Sloane, President, OEIS Foundation
>>> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
>>> > Phone: 732 828 6098; home page: http://NeilSloane.com
>>> > Email: njasloane at gmail.com
>>> >
>>> > _______________________________________________
>>> >
>>> > Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
--
Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com
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