Generalized Bell Numbers

[Franklin T. Adams-Watters]

I happened to look at new sequence A111579 today.  The description of this sequence needs to be clarified a bit; the Q should be replaced by Q(m,k), with the note that Q(m,k)=(k-1)*m+1.  The table is better described as a square array by antidiagonals, not as a triangle, and then a(m,n) is the sum of the terms in the nth row of the generalized Stirling triangle using Q(m,k) as the coefficients for the kth column.

Given this, the zeroth row of the array (this is the zeroth column from the original description) is wrong.  This row is given as all ones, which would correspond to a Q sequence of (0,0,0,...).  However, the sequence should be (1,0,-1,-2,...) to be consistent with the rest of the array.  A quick look at the columns of the square array (particularly the 2nd column, but every column is a polynomial except for the first row) shows that the values given do not fit.

Now it starts to get interesting.  When we build the sequence with (1,0,-1,-2,...), we get:
1,2,3,3,2,3,5,-4,5,55,-212,201,2381,-15350,35183,145359,-1821438,...
This sequence is not in the OEIS, but there is A080094:
1,-3,3,-1,3,-5,-2,-5,55,106,201,-2381,-7675,-35183,145359,910719,...
Offset by 1, this differs by a factor of (-1)^n, with an additional divisor of 2 every third term.  Looking at A080093-A080095, we see that in fact A080094(n) is being divided by 2^(n+1), except that every third term is only divided by 2^n.  And, of course, this is exactly the factor of 2 that distinguishes these terms.

"Restoring" this factor of 2 to A080093 gives us a match for A000296.

So, can anyone prove either or both of these correspondences (new sequence to A080094, or A000296 to A080093)?

Assuming that such proofs can be produced, I think we should make the following changes:
* Add the sequence above.  Cross reference it with A080094.
* Cross reference A000296 with A080093.
* Add terms 1, -1, 1 respectively as index 0 terms to A080093-A080095.  (Note that this matches the definition of these sequences: Sum 1/(2k+1)! = sinh(1) = e - 1/e.)
* Edit A111579 to remove (or fix) the first row, and fix the description.
-- 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645


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