# [seqfan] Re: updated list of formulas (conjectures)

Simon Plouffe simon.plouffe at gmail.com
Mon Feb 14 14:57:19 CET 2011

```
Hello,

lgdegf means that the exponential of the integral
of x+2 is the exponential generating function,

since we have that x + 2 - f(x) = 0 then

f(x) = x + 2

by doing exactly the inverse

exp(int(x+2,x));

we get the known generating function of that
sequence which is : exp(2*x + x^2/2) :

(now here is some Maple output) sorry if comes out
all scrambled : the page width is 75 and it
has to be read with COURRIER font or equivalent .

------------------------------------------------------------
> exp(int(x+2,x));
2
exp(2 x + 1/2 x )

> series(%,x,24);
2        3   43  4   71  5   499  6   185  7   7193   8
1 + 2 x + 5/2 x  + 7/3 x  + -- x  + -- x  + --- x  + --- x  + ----- x  +
24      60      720      504      40320

14593   9   17587   10    269039   11    486071   12    5658823    13
------ x  + ------ x   + -------- x   + -------- x   + ---------- x
181440      518400       19958400       95800320       3113510400

54229907    14    133453429    15     1347262321    16
+ ----------- x   + ------------ x   + -------------- x   +
87178291200       653837184000       20922789888000

99500491     17     36833528197     18      99518837527     19
------------- x   + ---------------- x   + ----------------- x   +
5081248972800       6402373705728000       60822550204416000

1097912385851     20      3088289136391      21
------------------- x   + -------------------- x   +
2432902008176640000       25545471085854720000

7081863329687      22       103351677649037      23      24
--------------------- x   + ----------------------- x   + O(x  )
224800145555521536000       12926008369442488320000

> seriestolist(%);

43  71  499  185  7193   14593   17587    269039
[1, 2, 5/2, 7/3, --, --, ---, ---, -----, ------, ------, --------,
24  60  720  504  40320  181440  518400  19958400

486071    5658823     54229907     133453429      1347262321
--------, ----------, -----------, ------------, --------------,
95800320  3113510400  87178291200  653837184000  20922789888000

99500491       36833528197        99518837527
-------------, ----------------, -----------------,
5081248972800  6402373705728000  60822550204416000

1097912385851        3088289136391          7081863329687
-------------------, --------------------, ---------------------,
2432902008176640000  25545471085854720000  224800145555521536000

103351677649037
-----------------------]
12926008369442488320000

> aa:=%:[seq(aa[k]*(k-1)!,k=1..nops(aa))];

[1, 2, 5, 14, 43, 142, 499, 1850, 7193, 29186, 123109, 538078, 2430355,

11317646, 54229907, 266906858, 1347262321, 6965034370, 36833528197,

199037675054, 1097912385851, 6176578272782, 35409316648435,

206703355298074]

Which is exactly the sequence.

That technique was used in my original article with
François Bergeron a long time ago and this simple idea
we had was the source of the GFUN program.
Here is the original article : François Bergeron
had the original idea of using standard calculus
techniques and Maple to get fancy generating functions,
it was the hell of a good idea!

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.em/1048610118

see all of the formulas here :

http://www.plouffe.fr/simon/OEIS/conjectures/OEIS_conjectured_formulas.txt

It explains the idea of using 'reverse engineering' on sequences
my using a simple Maple program.

I wish this explains more ?

Best regards et bon après-midi.

ps : we had a lot of fun doing this.

Simon Plouffe

```