some eta products

Sat May 3 14:40:29 CEST 2008

s=-log(prod(j=1,N,1-x^j)); \\ A000203 sum of divisors
s
s=serconvol(s,c)
s=-log(prod(j=1,N,(1-x^j)^(1/j))); \\ A000005 the number of divisors of n.
s=-log(prod(j=1,N,1-x^j)); \\ A000203 sum of divisors
s=-log(prod(j=1,N,(1-x^j)^j)); \\ A001157 sum of squares of divisors of n.
s=-log(prod(j=1,N,(1-x^j)^(j^2))); \\ A001158 sum of cubes of divisors of n. 
s=-log(prod(j=1,N,(1-x^j)^(j^3))); \\ A001158 sigma_4(n)
s=-log(prod(j=1,N,(1+x^j)^(1/j))); \\ A048272 Number of odd divisors of n minus number of even divisors of n.
s=-log(prod(j=1,N,(1+x^j)^(1))); \\ A000593   Sum of odd divisors of n. A115607
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Message-ID: <79c4b86c0805030646l346594a7kdf07845dadeb7ec8 at mail.gmail.com>
Date: Sat, 3 May 2008 22:46:39 +0900
From: "Thomas Copeland" <tccopeland at gmail.com>
To: seqfan at ext.jussieu.fr
Subject: Seeking an endorsement for arXiv.org
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Dear seqfans,

I've attached to this email a short didactic paper on some basic
relations between rooted trees and multivariable calculus that involve
several classical number triangles. I'm hoping to store it on the
arXiv.org site and link to it in the OEIS. Before I can do that, I
need an endorsement for the Combinatorics section of the archive. Can
someone help me with that? The paper is attached as a pdf file.

Sincerely,
Tom Copeland

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------=_Part_29434_10301294.1209822400565--

* Joerg Arndt <arndt at jjj.de> [May 03. 2008 14:54]:
> In pari/gp:
> 

> [...]

> 
> \\ A107742   G.f.: Product(Product(1+x^(i*j),i=1..infinity),j=1..infinity).
> 1/prod(j=0,N,eta(x^(2*j+1)));
> Vec(%)

==
prod(j=1,N,(1+x^j)^numdiv(j)) \\  A107742

> [...]

Some eta-LGF:

\\ A060640   If n = Product p_i^e_i then
\\   a(n) = Product (1 + 2*p_i + 3*p_i^2 + ... + (e_i+1)*p_i^e_i)
t=1/prod(j=1,N,eta(x^(j)))
t=log(t)
t=serconvol(t,c)
Vec(t)
\\ [1, 5, 7, 17, 11, 35, 15, 49, 34, 55, 23, 119, 27, 75, 77, 129, 35]

i.e. (LGF):  sum(A060640(n)*x^n/n) = - log( prod(j=1,N,eta(x^(j))) )

\\ A001001   Sublattices of index n in generic 3-dimensional lattice. 
t=1/prod(j=1,N,eta(x^(j))^j)
t=log(t)
t=serconvol(t,c)
Vec(t)
\\ [1, 7, 13, 35, 31, 91, 57, 155, 130, 217, 133, 455, 183, 399, 403, 651, 307]

\\ A109386   G.f. is the logarithm of the g.f. of A107742: ...
t=1/prod(j=0,N,eta(x^(2*j+1)))
t=log(t)
t=serconvol(t,c)
Vec(t)
\\  [1, 3, 7, 7, 11, 21, 15, 15, 34, 33, 23, 49, 27, 45, 77, 31, 35]

\\ A027847   Sum sigma(n/d)*d^3; d|n.
t=1/prod(j=1,N,eta(x^(j))^(j^2))
t=log(t)
t=serconvol(t,c)
Vec(t)
\\ [1, 11, 31, 95, 131, 341, 351, 775, 850, 1441, 1343, 2945, 2211, 3861, 4061, 6231, 4931]

\\ A007429   Inverse Moebius transform applied twice to natural numbers. 
t=1/prod(j=1,N,eta(x^(j))^(1/j))
t=log(t)
t=serconvol(t,c)
Vec(t)
\\ [1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19]

\\ A027848   Sum_{ d|n } sigma(n/d)*d^4.
t=1/prod(j=1,N,eta(x^(j))^(j^3))
t=log(t)
t=serconvol(t,c)
Vec(t)
\\ [1, 19, 85, 311, 631, 1615, 2409, 4991, 6898, 11989, 14653, 26435, 28575, 45771, 53635, 79887, 83539]

Hi,
firstly sending a big attachment to a mailinglist is not a good thing.
Put it on the web and send a URL.

A very quick look on your paper:
- define the symbols you use.
- the images seems to have suffered a jpeg stage,

cheers,   jj

* Thomas Copeland <tccopeland at gmail.com> [May 03. 2008 16:26]:
> Dear seqfans,
> 
> I've attached to this email a short didactic paper on some basic
> relations between rooted trees and multivariable calculus that involve
> several classical number triangles. I'm hoping to store it on the
> arXiv.org site and link to it in the OEIS. Before I can do that, I
> need an endorsement for the Combinatorics section of the archive. Can
> someone help me with that? The paper is attached as a pdf file.
> 
> Sincerely,
> Tom Copeland