[seqfan] Help needed with triangle related to new sequence

Ed Jeffery ed.jeffery at yahoo.com
Mon Nov 28 01:53:03 CET 2011


Seqfans,

I have a nice new sequence I am working on. It is read from the anti-diagonals of a table for which the rows are sequences based on certain unit-primitive matrices 

(https://oeis.org/wiki/User:L._Edson_Jeffery/UnitPrimitiveMatrix). I don't want to give the sequence here until I finish. However, it turns out that some of the columns of the table are core sequences already in OEIS, but all the columns have related generating functions of the form 

F_n(x)=P_n(x)/(1-x)^n, 

where P_n(x) is a polynomial in x. It turns out that the n-th polynomial in the numerator has (strictly non-negative) coefficients taken from the n-th row of the following triangle, letting n=0,1,2,... and k=0,1,...,n: 

T(n,k)=
{1                                      }
{1,  1                                  }
{1,  3,   1                             }
{1,  7,   7,    1                       }
{1, 14,  31,   14,    1                 }
{1, 26, 109,  109,   26,    1           }
{1, 46, 334,  623,  334,   46,   1      }
{1, 79, 937, 2951, 2951,  937,  79,  1  }
{1,133,2475,12331,20641,12331,2475,133,1}
etc.
The second column or diagonal of this triangle seems to be the sequence 

A001924={1,3,7,14,26,46,79,133,221,...} (https://oeis.org/A001924). 

The row sums of the triangle are evidently the sequence

A000111={1,2,5,16,61,272,1385,7936,50521,...} (https://oeis.org/A000111). I also can't find the anti-diagonal sums of this triangle. Has anyone seen this triangle before and, if so (or otherwise), could someone please let me know how to generate these coefficients?

Thanks,

Ed Jeffery



More information about the SeqFan mailing list