permutation transform?

[Ross La Haye]

Recently, I've become interested in the binomial transforms of integer
sequences and today at work I was wondering if a similar transform has been
defined that I'll call the permutation transform, and, if not, if it's
interesting.  Simply, the transform (and its inverse) would be the same as
with the binomial except that C(n,k) = n! / k!(n-k)! would be replaced with
P(n,k) = n! / (n-k)!.  So for instance, for

1, 1, 1, 1, 1, 1, 1, 1, 1...

the inverse permutation transform would give

1, 0, 1, 2, 9, 44, 265, 1854, 14833... (A000166) 

and the permutation transform would give 

1, 2, 5, 16, 65, 326, 1957, 13700, 109601 (A000522).

Additionally, so far I've found transforms and inverse transforms in OEIS
for the natural numbers, the triangular numbers, and the powers of 2, and
transforms but no inverse transforms for the even and odd numbers.  I
checked the transform and inverse transform for the primes, but didn't find
anything in OEIS.  Here are the first several terms respectively --

2, 5, 18, 83, 506, 3417, 26570, 225143, 2144658...
2, 1, 6, 19, 146, 573, 6134, 34879, 433026...

Conjecture: for each sequence the terms are alternatingly odd or even.  If
true, is this interesting...?

Ross


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