[seqfan] Superseeker question + two interesting patterns

[Johan de Ruiter]

Dear Seqfans,

I have a beginners question regarding the Superseeker.

For a sequence starting with
[2,16,128,1156,10952,107584,1083392,11115556,115702472], it returned
[4096 - 2048 a(n) + 384 a(n)^2 - 32 a(n)^3 + a(n)^4 , lgdegf]. That
polynomial happens to equal (a(n)-8)^4.
Does this mean that it suggested the generating function f(x) =
f'(x)/8, so f(x) = c * e^(8x) and hence the coefficients
of the exponential generating function should equal c * powers of 8?
This is not the case.
I know the response email warned it might only be an approximation,
but maybe I misunderstood something important?

---

Some context, for those who might be interested: I was wondering how
many configurations I could end up with when starting with a row of
piles of (say) playing cards and creating a new row of piles of
playing cards while only moving a card when it is currently on top of
a pile and moving each card exactly once. In particular I want to fix
the heights of the piles.

Example 1: From 2 piles of height 2, to 4 piles of height 1, there are
24 possibilities. (all permutations)
Example 2: From 4 piles of height 1, to 2 piles of height 2, there are
24 possibilities. (symmetrical with Example 1; this symmetry always
applies)
Example 3: From 2 piles of height 2, to 2 piles of height 2, there are
16 possibilities. (no directed cycles allowed)

When I computed a table of values for configurations with an equal
number of piles in the start and end position, all with the same
height, I noticed 2 remarkable patterns.

Pattern 1) For 2 piles of height h in both the start and end
configuration, the number of possible end configurations seems to be
either a square or twice a square, depending on h being odd or even. I
don't know why yet. Here are the first 19 terms:

2 = 2  * 1^2
16 = 4^2
128 = 2  * 8^2
1156 = 34^2
10952 = 2  * 74^2
107584 = 328^2
1083392 = 2  * 736^2
11115556 = 3334^2
115702472 = 2  * 7606^2
1218289216 = 34904^2
12948910592 = 2  * 80464^2
138708574096 = 372436^2
1495661223968 = 2  * 864772^2
16218468710656 = 4027216^2
176727219273728 = 2  * 9400192^2
1933956651447076 = 43976774^2
21243204576601928 = 2  * 103061158^2
234121111199439424 = 483860632^2
2587943032046002688 = 2 *  1137528688^2

Pattern 2) If all piles are of height 2, the number of possible
configurations for n piles is (2n)! - n^2(2n-2)! =
(3n-2)/(4n-2)*(2n)!.
I don't have an elegant interpretation of this elegant formula. I do
have a nice recursive definition, but I still need to formally prove
this is equivalent to the closed formula.

Two forms of the recursion are:
a_k = sum for t=0..k-1 of a_t*k!(k-1)!*(2^(2k-2t-1)-1)/(t!)^2
a_k = sum for t=0..k-1 of a_t(k!!(k-1)!!/(t!!)^2 - k!(k-1)!/(t!)^2)

Best regards,
Johan de Ruiter