"A dream" of a series :-)

Thu Jun 5 15:40:16 CEST 2008

Am 05.06.2008 15:13 schrieb David W. Wilson:
> Mr. helms effectively defines the sequence of rations c where
> 
>     e^x = PROD(n >= 1; 1 + c_n x^n)
> 
> Thus c = (1, 1/2, -1/3, 3/8, -1/5, ...), indexed starting at 1.
> 
> I observe that taking the log of this expression yields
> 
>     x = SUM(n >= 1; log(1 + c_n x^n))
> 
> If you take a Taylor series of the log expression and reorder the resulting
> sum, you can obtain an expression for c_n in terms of earlier c_d where d|n.
> 
This is a nice idea -
 thanks!

Gottfried


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Gottfried (and Seqfans),=20
A137852=20=20
G.f.: Product_{n>=3D1} (1 + a(n)*x^n/n!) =3D exp(x).=20
9001600, ...=20
COMMENT=20=20
Equals signed A006973 (except for initial term), where A006973 lists the=20
dimensions of representations by Witt vectors.=20

FORMULA=20=20
a(n) =3D (n-1)!*[(-1)^n + Sum_{d divides n, 1<d<n} d*( -a(d)/d! )^(n/d) ] f=
or n>1 with a(1)=3D1.=20
=20=20
Is that something like you were looking for?
=20=20
> > If you take a Taylor series of the log expression and reorder the resul=
ting
> > sum, you can obtain an expression for c_n in terms of earlier c_d where=
> >=20
> This is a nice idea -
> thanks!
>
> Gottfried
=20
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<html><P>Gottfried (and Seqfans), <BR>     See the form=
ula found in: <BR>A137852  <BR>G.f.: Product_{n>=3D1} (1 + a(n)*x^n=
/n!) =3D exp(x). <BR> 1, 1, -2, 9, -24, 130, -720, 8505, -35840, 41277=
6, -3628800, 42030450, -479001600, ... <BR>COMMENT  <BR>Equals si=
gned A006973 (except for initial term), where A006973 lists the <BR>dimensi=
ons of representations by Witt vectors. <BR><BR>FORMULA  <BR>a(n) =3D =
(n-1)!*[(-1)^n + Sum_{d divides n, 1<d<n} d*( -a(d)/d! )^(n/d) ] for =
n>1 with a(1)=3D1. <BR>  <BR>Is that something like you were lookin=
g for?<BR>     Paul<BR>  <BR>> > If you take=
t; sum, you can obtain an expression for c_n in terms of earlier c_d where =
d|n.<BR>> > <BR>> This is a nice idea -<BR>> thanks!<BR>><BR=
>> Gottfried<BR> </P></html>

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