The sequence [OEIS A056296](https://oeis.org/A056296) can be obtained using


$
 a(n)={1\over n}\sum\_{d\backslash n}\varphi(d)\begin{cases}
 {n/d+2\brace3}-{n/d+1\brace3}, & \text{$6\backslash d$;} \\
 {n/d+2\brace3}-3{n/d+1\brace3}+3{n/d\brace3}, & \text{$3\backslash d$;} \\
 2{n/d+1\brace3}-2{n/d\brace3}, & \text{$2\backslash d$;} \\
 {n/d\brace3}, & \text{else.} \\
 \end{cases}
$


or replacing the Stirling subset numbers


$
 a(n)={1\over n}\sum\_{d\backslash n}\varphi(d)\begin{cases}
 3^{n/d}-2^{n/d}, & \text{$6\backslash d$;} \\
 \left(3^{n/d}-2^{n/d}+1\right)/2, & \text{$3\backslash d$;} \\
 \left(2\cdot3^{n/d}-3\cdot2^{n/d}\right)/3, & \text{$2\backslash d$;} \\
 \left(3^{n/d}-3\cdot2^{n/d}+3\right)/6, & \text{else.} \\
 \end{cases}
$


Is there an ordinary generating function for this sequence?



Ordinary Generating Function for OEIS A056296?
It appears that


$ \sum\_{d>0}{\varphi(d)\over6d}\begin{cases}
 -6\log(1-3x^d)+6\log(1-2x^d), & \text{$6\backslash d$;} \\
 -3\log(1-3x^d)+3\log(1-2x^d)-3\log(1-x^d), & \text{$3\backslash d$;} \\
 -4\log(1-3x^d)+6\log(1-2x^d), & \text{$2\backslash d$;} \\
 -\log(1-3x^d)+3\log(1-2x^d)-3\log(1-x^d), & \text{else.} \\
 \end{cases}
$


works. Kociemba's o.g.f. for [OEIS A000013](https://oeis.org/A000013) was very helpful to me.


