Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)).
(history;
published version)
COMMENTS
The Akiyama-Tanigawa algorithm applied to a sequence yields the same result as the Stirling-Bernoulli Transform applied to the same sequence. See Philippe Deléham's comment of Nov 05 2011May 26 2015. - Shel Kaphan, May 16 2024
Discussion
Fri May 24
17:22
Shel Kaphan: I just changed the date of previous comment that I referred to. There were two comments by the same person and I had referenced the wrong one previously.
Discussion
Thu May 16
16:49
Michel Marcus: so comment does not really apply to this sequence ?
17:13
Shel Kaphan: I would think it applies since according to the previous comment I quoted this sequence effectively defines the Stirling-Bernoulli transform. I can't imagine another sequence that would be more relevant.
COMMENTS
The Akiyama-Tanigawa algorithm applied to a sequence yields the same result as the Stirling-Bernoulli Transform applied to the same sequence. See Philippe Deléham's comment of Nov 05 2011. - Shel Kaphan, May 16 2024
COMMENTS
Row sums for n > 0 are zero. - Shel Kaphan, May 14 2024
Discussion
Sun May 12
02:13
Kevin Ryde: Program code should be a 1 liner def T(i,j); return etc. There's no merit in inefficient code for a sequence this easy. (Yes present the mathematics, but program code is not mathematics.)
08:01
Shel Kaphan: Sorry, what's "inefficient"? Some things don't fit on one line. Anyway ask Peter Luschny whose structure this is, and who has used it in a few places. If you are referring to separating out the function being transformed from the transform, that is not "inefficient", it is for clarity and there *is* merit in that in program code. It doesn't affect the running time in any meaningful way. And as always, if editors don't want something, simply revert.
08:02
Shel Kaphan: Or if you are referring to demonstrating that the AT transform produces the table, I think there's pedagogical merit, but if you don't want it, please just toss it.
17:33
Peter Luschny: Shel, your comment is fundamentally correct, but each case should be considered individually. In this case, I think Kevin's objection is justified, especially because there is a more efficient recursion here (see Zumkeller), and I am therefore deleting the program but leaving the comment.
COMMENTS
The n-th diagonal consists of n^k. This can also be generated as the Akiyama-Tanigawa algorithm applied to the sequence binomial(n+k,k), k >= 0. See Python program below. - Shel Kaphan, May 03 2024
PROG
for n in range(1, 8): print([n], ATtransform(n, 8, f)) # - Shel Kaphan, May 03 2024.
