[seqfan] Re: "A dream" of a series :-)

[Maximilian Hasler]

PS: Also, you might have cited the sequence
A137852 : Product_{n>=1} (1 + a(n)*x^n/n!) = exp(x).
1, 1, -2, 9, -24, 130, -720, 8505, -35840, 412776, -3628800,...

It is the sequence of your coefficients multiplied by n! as to have an
integer sequence:

gh1(n)={e=exp(x+O(x^n)); vector( n-1,i, c=polcoeff(e,i); e/=(1+x^i*c); c)}
gh1(15)
%4 = [1, 1/2, -1/3, 3/8, -1/5, 13/72, -1/7, 27/128, -8/81, 91/800,
-1/11, 1213/13824, -1/13, 505/6272]

gh(n)={e=exp(x+O(x^n)); vector( n-1,i, c=polcoeff(e,i); e/=(1+x^i*c); c*i!)}
gh(15)
%5 = [1, 1, -2, 9, -24, 130, -720, 8505, -35840, 412776, -3628800,
42030450, -479001600, 7019298000]

Regards,
Maximilian

On Sun, Feb 15, 2009 at 9:07 AM, Maximilian Hasler
<maximilian.hasler at gmail.com> wrote:
> Very nice article.
> Just, for the next update, you may want to fix the numerical data at
> the very beginning of the article:
>
> exp(x+O(x^9))
> %1 = 1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 +
> 1/5040*x^7 + 1/40320*x^8 + O(x^9)
>
> %/(1+x)
> %2 = 1 + 1/2*x^2 - 1/3*x^3 + 3/8*x^4 - 11/30*x^5 + 53/144*x^6 -
> 103/280*x^7 + 2119/5760*x^8 + O(x^9)
>
> %/(1+x^2/2)
> %3 = 1 - 1/3*x^3 + 3/8*x^4 - 1/5*x^5 + 13/72*x^6 - 15/56*x^7 +
> 533/1920*x^8 + O(x^9)
>
> and not -3/8 x^3 as written in the article.
> Regards,
> Maximilian
>
> On Sun, Feb 15, 2009 at 3:16 AM, Gottfried Helms
> <Annette.Warlich at t-online.de> wrote:
>> Dear seqfans -
>>
>>  there was no time in summer, when I discussed this
>>  "dream of a series". In the meantime I could put things
>>  together into a short readable article.
>>  Hope you enjoy!
>>
>>    http://go.helms-net.de/math/musings/dreamofasequence.pdf
>>
>>  The main sequence exists in OEIS (A067911, (*1)) although
>>
>>  - not with the relation to the generating process as described
>>     in the article (I'll supply that information soon)
>>
>>  - modified in the sense, that in my article the sequence
>>     is defined by denominators of rational numbers and some
>>     missing primefactors may be seen as cancelled by the numerators
>>     of the coefficients by the rational-arithmetic-system in Pari/GP.
>>
>>  If I want to add the numerator-sequence to OEIS we must decide,
>>  whether I expand my numerators such that the denominators match
>>  the values in A067911 or whether I should send numerators and
>>  denominators in their reduced versions.
>>  Neill - what's your opinion?
>>
>> Gottfried Helms
>>
>>
>> (*1)  http://www.research.att.com/~njas/sequences/A067911
>>
>>
>>
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>