[seqfan] Re: A new triangle revisited
Ed Jeffery
ed.jeffery at yahoo.com
Thu Dec 8 00:39:17 CET 2011
Seqfans,
I am replying to my own post, but this is an update on the triangle I found:
1
1,1
1,3,1
1,7,7,1
etc.
I conjectured that the generating functions for the rows of this triangle are convolutions with a specific pattern. This is evidently false, since the idea has now failed for the fourth column, so it is just a coincidence that the first three generating functions followed that pattern. I still wonder if the generating functions are convolutions of polynomials with coefficients given in some way by A187660, but it is going to be very tough to determine the g.f. for even the fourth column at this point. I have been working on that one for a couple of days now, with no result. This leads me to believe that I was lucky to work out the g.f. for the third column.
I'll add another conjecture here, namely, that the m-th row of the triangle is palindromic for all m, that is, one gets the same sequence of row entries by reading them in reverse order (as with the triangle of binomial ciefficients). This is, so far, supported by verification through (albeit only) row 19.
Ed Jeffery
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