# [seqfan] Re: sigma(4n+1) mod 4 and partitions

Richard Mathar mathar at strw.leidenuniv.nl
Thu Mar 24 16:33:46 CET 2011

```http://list.seqfan.eu/pipermail/seqfan/2011-March/007365.html says

gg> Gjergji Zaimi proved [1]:
gg> A001935(n) mod 4 = sigma(8n+1) mod 4
gg> and
gg> A001936(n) mod 4 = sigma(4n+1) mod 4
gg>
gg> A001935 Number of partitions with no even part repeated
gg> A001936 Expansion of q^(-1/4) (eta(q^4) / eta(q))^2 in powers of q

There is another interpretation which may help to find a inductive
proof of these variants: A001935(n) is the number of ways to partition
n into a set of distinct parts plus a set of distinct even parts.
[The equivalent arithmetic recurrences is the convolution of A000009
already mentioned in the sequence, and the observation is simply another
look at the g.f product_{j>=1} (1+x^j)*(1+x^(2j).] For example a(9) = 22
counts the 22 partitions
= (1+2)+(2+4) = (1+2+4)+(2)
=(2+3+4)+() = (3+4)+(2) = (3+2)+(4) = (3)+(2+4)
= (1+4)+(4)
= (2+5)+(2)
= (1+3+5)+()
= (4+5) = (5)+(4)
= (1+2+6)+() = (1+2)+(6) = (1+6)+(2) = (1)+(2+6)
= (3+6)+() = (3)+(6)
= (2+7)+() = (7)+(2)
= (1+8)+() = (1)+(8)
= (9)+()
where the first () embraces the distinct integers and the second ()
embraces the distinct even integers.

sigma(4n+1) is A112610(n). It might be useful to look at the representation
of A112610 as convolution of  psi(q)^2*phi(q)^2 as found in the
Hirschhorn reference, convolution of A004018 and A008441,
and then look (mod 4) at each of the factors.
RJM

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