[seqfan] A new triangle revisited

[Ed Jeffery]

Seqfans,

On Nov 28, I posted a message about a triangle associated with what 

I thought was a new sequence read from the anti-diagonals of a table. 

After ignoring the first column it turned out that my table was the same 

as A050447 (https://oeis.org/A050447) and its transposed version the 

same as A050446 (https://oeis.org/A050446). The table A050447 is

1,1,1,1,1,1,1,... 
1,2,3,4,5,6,7,... 
1,3,6,10,15,21,28,... 
1,5,14,30,55,91,140,... 
1,8,31,85,190,371,658,...

I added a comment to A050447 stating some conjectures (which I need 

to correct), a few of whichare the following.

Rows of the table:


(iii) The n-th row (n=1,2,...) of the table has generating function of the form
f_n(x) = B_n(x)/(1-x)^n,
in which B_1(x) = B_2(x) = 1 and, for n >= 3, B_n(x) is a polynomial of 

degree n-3 in x of the form
B_n(x) = b_0 + b_1*x + ... + b_(n-4)*x^(n-4) + b_(n-3)*x^(n-3)
in which the coefficients b_0, ..., b_(n-3) are given by the (n-2)-th row of 

the triangle

T=
1
1,1
1,3,1
1,7,7,1
1,14,31,14,1
1,26,109,109,26,1
etc.

This triangle definitely is not in the OEIS database, and I will submit it 

sometime after the holidays since its definition still needs a lot of work. 

As an example of its utility, row 8 of the table is the sequence
{1,34,464,2037,8272,26585,72302,173502,377739,...}.
By (iii), f_8(x)=B_8(x)/(1-x)^8, and using row 8-2=6 of the triangle we have 

B_8(x)=1+26*x+109*x2+109*x^3+26*x^4+x^5.
Therefore the generating function for row 8 of the table is
f_8(x)=(1+26*x+109*x2+109*x^3+26*x^4+x^5)/(1-x)^8.

Columns of the triangle:


(iv) The second column of the triangle is A001924 shifted to the right one 

place, so this column has generating function = x/((1-x-x^2)*(1-x)^2).

(v) However, subsequent columns of the triangle are unknown. In particular, 

column 3, which is the sequence {1,7,31,109,...}, has generating function = 

x^2*(1-x^2-x^3-x^4+x^5)/((1-2^x-x^2+x^3)*(1-x-x^2)^2*(1-x)^3).(This sequence 

I will submit to OEIS as well.)

I can't help but suspect a certain pattern in the denominators of the 

generating functions; therefore:

(vi) Generally, for the n-th column of the triangle, the generating function 

will be a convolution of the form
x^(n-1)*A_n(x)/(P_n(x)*[P_(n-1)(x)]^2*...*[P_2(x)]^(n-1)*[P_1(x)]^n,
in which A_n(x) and the P_j(x), j in {1,2,...,n}, are polynomials in x, with 

P_j(x) being monic, of degree j, and of the form
P_j(x)=c_0*x^j+c_1*x^(j-1)+...+c_(j-1)*x+c_j
in which the coefficients c_0,...,c_j are given by row j of A187660.

What I really need are more terms in the columns of my triangle so I can 

at least verify the first few generating functions or, perhaps, discard the 

whole idea if I fail. The difficulty for me is that my software can only handle 

15-digit numbersin arrays. 


Would someone please help me get more terms? 


I suspect that from the convolution for the n-th column arises a recurrence 

relation requiring knowledge of at most the first s(n) terms (the initial 

conditions), where s={1,4,10,20,35,...}=A000292. So, already for column 

four. I need more terms.

Finally, I have an idea of how the columns can be generated, if someone 

would like to help me.

Thanks,

Ed Jeffery