odd, funny, peculiar ...

[Jon Awbrey]

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Because (-2)/(1-(-2)) = -(2/(1-2))  ...

Seriously, folks, this seems to be dimly related --
I supply the dim -- to an old problem that I have
sometimes puzzled over.  Here's the beginnings of
an exposition of what I don't understand so far:

http://forum.wolframscience.com/showthread.php?s=&threadid=285

Under conditions where the formal manipulations make sense --
not a big enough where -- so let's work purely formally
at first and worry about how to fix things later ...

Going from A(x) = a_0 + a_1 x + a_2 x^2 + ..., that counts some species !A!

to 1/(1 - A(x)) = 1 + A(x) + A(x)^2 + A(x)^3 + ...

is formally associated with picking 0-tuples, 1-tuples, 2-tuples, ... from !A!

For example, if U(x) = x = 0*x^0 + 1*x^1 + 0*x^2 + 0*x^3 + ...
counts a species !U! where we have a single object of weight 1,
say, !U! = {1},

then V(x) = U(x)NAD = 1/(1 - U(x)) = 1 + U(x) + U(x)^2 + ... = 1 + x + x^2 + ...
counts a species !V! where we have a unique object of each non-negative weight,
analogous to the k-tuples (), (1), (1,1), (1,1,1), ... ~=~ 0, 1, 2, 3, ...

and W(x) = xV(x) = xU(x)NAD = x/(1-x) = x + x^2 + x^3 + ... counts a species !W!
where we have a unique object of each positive weight, say, !W! = {1, 2, 3, ...),
typically associated with adding a root to the species counted by V(x).

Huh, I seem to see further than I did last time,
so I'll leave it as an exercise for the writer ...

Jon Awbrey

> Paul D. Hanna wrote:
> 
> Seqfans,
>         Consider the peculiar generating function:
> 
> (*)  x/(1-x) = Sum_{n>=1} (1/n!)*Product_{j=0..n-1} A(2^j*x)
> e.g.,
> x/(1-x) = A(x) + A(x)*A(2*x)/2! + A(x)*A(2*x)*A(2^2*x)/3!
>            + A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4!
>            + A(x)*A(2*x)*A(2^2*x)*A(2^3*x)*A(2^4*x)/5! + ...
> 
> A solution, A(x), is the e.g.f. of sequence A111814:
> A(x) = x - 2*x^3/3! + 216*x^5/5! - 568464*x^7/7! + 36058658688*x^9/9!
>               - 53694310935340800*x^11/11! +-...
> 
> It appears that A(x) is an odd function, but for no compelling reason.
> 
> Can anyone shed light on the mystery as to
> why A(-x) = -A(x) would be a necessary consequence of (*)?
> 
> The function A(x) is very interesting since the following function G(x):
> (**)  G(x) = Sum_{n>=1} (m*2^k)^n/n!*Product_{j=0..n-1} A(2^j*x)
> is always an integer series for all integer m and integer k>=0, and
> G(x) equals the g.f. of column k of the matrix m-th power of triangle A078121.
> This (**) is a result of an observation made by Gottfried Helms over 2 years ago!
> 
> Thanks,
>       Paul

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