# [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

```