Yet more multiplicatives

[David Wilson]

I'm sporadically working on Mitchell Harris's project to properly annotate 
as many as possible of the multiplicative sequences in the OEIS.

Right now I'm working on A030199:

%I A030199
%S A030199 
0,1,1,1,0,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,1,1,1,1,0,1,0,1,0,0,0,
%T A030199 
0,0,0,0,1,0,1,0,0,0,0,1,1,1,1,1,0,0,0,1,0,0,0,1,2,0,0,1,0,1,0,0,0,0,1,
%U A030199 
0,1,0,1,0,1,0,0,1,0,0,1,1,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,1,0,0,2,0,0,1
%V A030199 
0,1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,1,-1,1,1,1,0,-1,0,-1,0,0,0,0,0,0,0,
%W A030199 
1,0,-1,0,0,0,0,-1,-1,1,1,-1,0,0,0,-1,0,0,0,1,2,0,0,1,0,1,0,0,0,0,-1,0,-1,0,-1,0,-1,0,
%X A030199 0,-1,0,0,-1,1,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,-1,0,0,2,0,0,-1
%N A030199 G.f.: x*Product( (1-x^k)*(1-x^(23*k)),k=1..infinity).
%C A030199 Expansion of eta(q)*eta(q^23).
[lines omitted]
%K A030199 sign
%O A030199 0,60
%A A030199 njas

Using the %N line definition, I extended the sequence to 1000 terms and 
verified it was multiplicative up to that point, so I am guessing it is 
indeed multiplicative.

I was wondering if multiplicativity might be apparent from the definition to 
someone better versed in these things.

I also noticed the %C line gives an "eta" expansion that I do not 
understand.  There are several empirically multiplicative sequences in 
Harris's list which have similar expansions, such as:

      A004530  ??  Expansion of (theta_2 + theta_3 + theta_4) / 2.

      A089810  ??  Expansion of Jacobi theta function 
(3theta_4(q^9)-theta_4(q))/2.

      A030201  ??  Expansion of eta(q^3)*eta(q^21).

      A094247  ??  Expansion of eta(q)eta(q^10)^3/(eta(q^2)eta(q^5)) in 
powers of q.

      A030187  ??  Expansion of eta(q)*eta(q^2)*eta(q^7)*eta(q^14).

      A030207  ??  Expansion of eta(q)^2*eta(q^2)*eta(q^4)*eta(q^8)^2.

      A030209  ??  Expansion of (eta(q)*eta(q^2)*eta(q^3)*eta(q^6))^2.

      A030210  ??  Expansion of (eta(q)*eta(q^5))^4.


And I have to wonder if the right person would be able to tell whether or 
not these functions are multiplicative.



- David W. Wilson

"Truth is just truth -- You can't have opinions about the truth."
   - Peter Schickele, from P.D.Q. Bach's oratorio "The Seasonings"