[seqfan] Formulas needed for new sequence.

[Ed Jeffery]

Fellow sequence fanatics,

I recently submitted the following seemingly trivial sequence to OEIS:

A191747={1,1,0,0,1,1,0,0,0,1,0,0,0,1,...} (offset=1). 

This sequence is just a concatenation of row entries of the N X N identity matrices and hasn't been reviewed yet. 

However, of interest is the sequence of all m in {1,2,3,...} for which A191747(m)=1. This definition gives the sequence 

A191748={1,2,5,6,10,14,...}, 

which can be read by anti-diagonals from the table 

1,  5,  14,  30,  55,  ..
2,  10, 25,  49,  84,  ..
6,  20, 43,  77,  124, ..
15, 37, 70,  116, 177, ..
31, 63, 108, 168, 245, ..
etc..

I haven't submitted this sequence yet, since I am in the midst of some adversity and need help with a formula for the general term: I haven't been able to concentrate long enough to get this done myself, so if some of you would like to help, it would be greatly appreciated.

Some analysis:

The first row of the table is the core sequence A000330, omitting the initial 0, and the first column is A056520. The j-th row can be found from the generating functions 

(1)  (-j+(2*j+1)*x-(j-1)*x^2)/(1-x)^4, j in {0,1,2,...},

by taking from the (j+1)-th term on, but this contradicts the definition of A000330 (corresponding to j=0) which has the initial term set to zero, although the generating functions are in agreement: the reason for the initial zero evidently is the alternative definition 

A056520(n)=A000300(n)+1.

Here, j=1 in (1) gives essentially A058373, ignoring the two initial zeros. (A058373, by the way, was defined with offset=-2 which is bewildering to me, and perhaps it should be changed to offset=0 after dropping the two zeros.) Putting j=2 gives -A058372 which leads me to believe that the signs should be reversed, that is, A058372 should be negated and rewritten in the database. As far as I can determine, none of the other rows or columns of the table appear in the OEIS database.

Continuing with analysis of the above table, similarly, the k-th column can be found from the generating functions

(2*k+1-(5*k+2)*x+4*(k+1)*x^2-(k+1)*x^3)/(1-x)^4, k in {0,1,2,...},

by taking the k-th term on.

Likewise, for j and k as above, for the j-th row R_j, we have the closed-form expression

R_j(n)=(n+1)*(2*n^2+n-6*j)/6, n=j+1,j+2,j+3,...;

and for the k-th column C_k,

C_k(n)=(n+2)*(2*n^2-n+6*k+3), n=j,j+1,j+2,....

In conclusion, I see no way of defining a generating function for this sequence, even though the rows and columns have generating functions with a common denominator, namely (1-x)^4. Maybe you all would like to try to find this function, if it exists. Furthermore, I suspect that a closed-form expression for a general term of the sequence exists, but I haven't been able to work it out. At present, my alleged mind can't handle complications with multiple indices (j and k) and varying offsets. So, I'll leave it to those with more experience and hope to get a unified expression soon, at which point I'll go ahead and submit the sequence.

Thanks in advance to everyone,

Ed Jeffery