[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
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