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A295495
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Number of dissections of an n-gon by nonintersecting diagonals into polygons with a prime number of sides counted up to rotations.
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5
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1, 1, 2, 5, 11, 36, 114, 410, 1458, 5488, 20786, 80770, 317378, 1265139, 5094139, 20718347, 84961256, 351086326, 1460591637, 6113826319, 25733864299, 108867782794, 462707558813, 1974991841442, 8463121111860, 36397780088126, 157066702354947, 679917566925030
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OFFSET
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3,3
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LINKS
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MATHEMATICA
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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]];
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PROG
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(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]))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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