On 3-regular partitions with odd parts distinct

We consider (text {pod}_3(n)), the number of 3-regular partitions with odd parts distinct, whose generating function is

$$begin{aligned} sum _{nge 0}text {pod}_3(n)q^n=frac{(-q;q^2)_infty (q^6;q^6)_infty }{(q^2;q^2)_infty (-q^3;q^3)_infty }=frac{psi (-q^3)}{psi (-q)}, end{aligned}$$

where

$$begin{aligned} psi (q)=sum _{nge 0}q^{(n^2+n)/2}=sum _{-infty }^infty q^{2n^2+n}. end{aligned}$$

For each (alpha >0), we obtain the generating function for

$$begin{aligned} sum _{nge 0}text {pod}_3left( 3^{alpha }n+delta _alpha right) q^n, end{aligned}$$

where (4delta _alpha equiv {-1}pmod {3^{alpha }}) if (alpha ) is even, (4delta _alpha equiv {-1}pmod {3^{alpha +1}}) if (alpha ) is odd.

We show that the sequence {(text {pod}_3(n))} satisfies the internal congruences

$$begin{aligned} text {pod}_3(9n+2)equiv text {pod}_3(n)pmod 9, end{aligned}$$

(0.1)

$$begin{aligned} text {pod}_3(27n+20)equiv text {pod}_3(3n+2)pmod {27} end{aligned}$$

(0.2)

and

$$begin{aligned} text {pod}_3(243n+182)equiv text {pod}_3(27n+20)pmod {81}. end{aligned}$$

(0.3)