[seqfan] Re: GF for A147843?

[Joerg Arndt]

* Joerg Arndt <arndt at jjj.de> [Mar 15. 2010 15:07]:
> (PARI/GP):
> deriv(-eta(x))
> 1 + 2*x - 5*x^4 - 7*x^6 + 12*x^11 + ...
> 
> here:
>  eta(x) = prod(n=1, \infty, 1-x^n)
> 
> I get
>  d/dx eta(x) = eta(x) / x * sum(n=1,\infty, -x^(n-1)/(1-x^n))
> this appears to check with the comment
> 
>   Convolved with the partition numbers A000041 = sigma(n) prefaced
>   with an 0: (0, 1, 3, 4, 7, 6, 12, 8, 15, 13,...).
> 

So a GF for A000203 is:
? -x*deriv(eta(x))/eta(x)
x + 3*x^2 + 4*x^3 + 7*x^4 + 6*x^5 + ...

Prog:
? Vec(-x*deriv(eta(x))/eta(x))
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, ...]