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The sequence OEIS 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?

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  • $\begingroup$ Is $d\backslash n$ a common notation for $d|n$ nowadays? $\endgroup$
    – Gerald Edgar
    Commented Mar 5, 2018 at 21:34
  • $\begingroup$ @Gerald, I think d\n is the notation I learned at the Ross program 50 years ago. But I haven't seen it much since. $\endgroup$
    – Gerry Myerson
    Commented Mar 5, 2018 at 21:57
  • $\begingroup$ Knuth uses $d\backslash n$ in Concrete Mathematics. $\endgroup$
    – Robert A. Russell
    Commented Mar 5, 2018 at 21:58
  • $\begingroup$ A Dirichlet generating function might be a better idea here. $\endgroup$
    – Alexander Burstein
    Commented Mar 6, 2018 at 5:14
  • $\begingroup$ The cases in the formulae are not mutually exclusive. $\endgroup$
    – Max Alekseyev
    Commented Mar 7, 2018 at 22:25

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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 was very helpful to me.

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