"Expansion of sin(ln(1+x))", "Order of alternating group A_n", and "rok symmetry"

[creigh at o2online.de]

Dear Seqfans, 

Perhaps it is possible for someone to find a formula for some or all of the 
les/les sequences for + .5'ii' + .5'jj' - .5'kk' + .5e  and + .5'ii' + .
5'jj' + .5'kk' - .5e, below.  These sequences should be interesting (I only 
tried one of these with Superseeker, which returned nothing). In each case, 
"ves" is 
http://www.research.att.com/projects/OEIS?Anum=A001710
a(n) = n!/2 for n >= 2. 


At the bottom of the page, I ask a similar question concerning the sequences "Expansion of 
cos(ln(1+x))" and "Expansion of sin(ln(1+x))". 
(it also makes use of "rok symmetry")

Note: the internet version of "FAMP" does not yet include the "rok symmetry". I 
will probably add it tomorrow (though I am still not completely sure if 
the definition of "rok" should stay the way it is or be slightly modified-
  I will have to test more examples to find out!). 

Happy New Year, 
Creighton 

Results for the floretion:  J*H
 

********************************************************** 

********************************************************** 
 + .5'ii' + .5'jj' - .5'kk' + .5e 
 

********************************************************** 

********************************************************** 
J =  + .5'ii' + .5'jj' - .5'kk' + .5e
H =  + e

  
Note: All sequences refer to the floretion J*H 

 
1vesrokseq: 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 199584, 2395008, 31135104, 
435891456, 653837184, 10461394944
 
 2tesrokseq: 1, 4, 17, 87, 528, 3720, 29880, 269640, 2701440, 2975616, 3574368, 
46503072, 651442176, 9776422656, 156485032704
 
 2lesrokseq: 1, 2, 7, 33, 192, 1320, 10440, 93240, 927360, 1016064, 1215648, 15767136, 
220340736, 3300321024, 52742866176, 89575694208
 
 1jesrokseq: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 


Results for the floretion:  E*D
 

**************************************************************************************************
 
 + .5'ii' + .5'jj' + .5'kk' - .5e 
 

**************************************************************************************************
 
E =  + .5'ii' + .5'jj' + .5'kk' - .5e
D =  + e

  
Note: All sequences refer to the floretion E*D 

 
1vesrokseq: 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 199584, 2395008, 31135104, 
435891456, 653837184, 10461394944
 
 2tesrokseq: -1, 0, 3, 21, 144, 1080, 9000, 83160, 846720, 9434880,
 1143072, 149688, 210760704, 3175780608, 50999300352, 86960345472
 
 2lesrokseq: 3, 6, 21, 99, 576, 3960, 31320, 279720, 2782080, 3048192, 3646944, 
47301408, 661022208, 9900963072, 158228598528
 
 1jesrokseq: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0


Results for the floretion:  K*J
 ******************************************************
 'ii' 
 
********************************************************** 

 
K =  + .5'ii' - .5'jj' + .5'kk' + .5e
J =  + .5'ii' + .5'jj' - .5'kk' + .5e

  
Note: All sequences refer to the floretion K*J 

 
1vesseq: 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 199584, 2395008, 31135104, 435891456, 
653837184, 10461394944
 
 1tesseq: 0, 3, 0, 60, 0, 2520, 0, 181440, 0, 199584, 0, 31135104, 0, 653837184, 
0
 
 1lesseq: 1, 0, 12, 0, 360, 0, 20160, 0, 1814400, 0, 2395008, 0, 435891456, 0, 10461394944,



Expansion of cos(ln(1+x))" and "Expansion of sin(ln(1+x))". 
Results for the floretion:  I*H
 

**************************************************************************************************

 
 + .5'i + .5'j + .5i' + .5j' 

1vesrokseq: 2, 3, 5, 10, 20, 10, -410, -5700, -65000, -734500, -8625500, -106951, 
-1407562, -19681155, -292105645, -459387617
 
cos(ln(1+x))
 1tesrokseq: 0, -1, -3, -10, -40, -190, -1050, -6620, -46800, -365300, -3103100, 
-282698, -2716272, -2691559, -26495469, -238131478, 
 
 1lesrokseq: 0, 2, 6, 20, 80, 380, 2100, 13240, 93600, 730600, 6206200, 565396, 
5432544, 5383118, 52990938, 476262956, 2788199648
 
Expansion of sin(ln(1+x)):
 1jesleftrokseq: 1, 1, 1, 0, -10, -90, -730, -6160, -55900, -549900, -5864300, 
-676104, -8395946, -11186357, -159300557, -2416003824, -38894192662
 
1jesrightrokseq = 1jesleftrokseq;

identity: ves = tes + les + jesright + jesleft