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

creigh at o2online.de creigh at o2online.de
Mon Jan 3 03:35:06 CET 2005

```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

```