[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
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