OFFSET
3,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 3..200
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
MATHEMATICA
DissectionsModCyclic[v_] :=
Module[{n = Length[v], q, vars, u, p}, q = Table[0, {n}]; q[[1]] = InverseSeries[x - Sum[x^i v[[i]], {i, 3, Length[v]}]/x + O[x]^(n+1)]; For[i = 2, i <= n, i++, q[[i]] = q[[i-1]] q[[1]]]; vars = Variables[q[[1]]]; u[m_, r_] := Normal[(q[[r]] + O[x]^(Quotient[n, m] + 1))] /. Thread[vars -> vars^m]; p = O[x]^n + x u[1, 1] - x^2 + (u[2, 1] - u[1, 2])/2 + Sum[v[[i]] Sum[EulerPhi[d] u[d, i/d]/i, {d, Divisors[i]}], {i, 3, Length[v]}]; Drop[CoefficientList[p, x], 3]];
DissectionsModCyclic[Boole[PrimeQ[#]]& /@ Range[1, 31]] (* Jean-François Alcover, Sep 26 2019, after Andrew Howroyd *)
PROG
(PARI) \\ number of dissections into parts defined by set.
DissectionsModCyclic(v)={my(n=#v);
my(q=vector(n)); q[1]=serreverse(x-sum(i=3, #v, x^i*v[i])/x + O(x*x^n));
for(i=2, n, q[i]=q[i-1]*q[1]);
my(vars=variables(q[1]));
my(u(m, r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t->t^m, vars)));
my(p=O(x*x^n) + x*u(1, 1) - x^2 + (u(2, 1)-u(1, 2))/2 + sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d, i/d))/i)));
vector(n, i, polcoeff(p, i))}
DissectionsModCyclic(apply(i->isprime(i), [1..30]))
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Nov 22 2017
STATUS
approved