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A295419
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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 and reflections.
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12
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1, 1, 2, 4, 8, 23, 64, 222, 752, 2805, 10475, 40614, 158994, 633456, 2548241, 10362685, 42485242, 175557329, 730314350, 3056971164, 12867007761, 54434131848, 231354091945, 987496927875, 4231561861914, 18198894300129, 78533356685275, 339958801585826
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OFFSET
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3,3
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COMMENTS
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a(n) first differs from A290816(n) at n=9 since this sequence does not allow the trivial dissection of a nonagon into a single nonagon.
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LINKS
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MATHEMATICA
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DissectionsModDihedral[v_] := Module[{n = Length[v], q, vars, u, R, Q, T, 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]; R = Sum[v[[2i+1]] u[2, i], {i, 1, (Length[v]-1)/2 // Floor}]; Q = Sum[v[[2i]] u[2, i-1], {i, 2, Length[v]/2 // Floor}]; T = Sum[v[[i]] Sum[EulerPhi[d] u[d, i/d], {d, Divisors[i]}]/i, {i, 3, Length[v]}]; p = O[x]^n - x^2 + (x u[1, 1] + u[2, 1] + (Q u[2, 1] - u[1, 2] + (x+R)^2/(1-Q))/2 + T)/2; Drop[ CoefficientList[p, x], 3]];
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PROG
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(PARI) \\ number of dissections into parts defined by set.
DissectionsModDihedral(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(R=sum(i=1, (#v-1)\2, v[2*i+1]*u(2, i)), Q=sum(i=2, #v\2, v[2*i]*u(2, i-1)), T=sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d, i/d))/i)));
my(p=O(x*x^n) - x^2 + (x*u(1, 1) + u(2, 1) + (Q*u(2, 1) - u(1, 2) + (x+R)^2/(1-Q))/2 + T)/2);
vector(n, i, polcoeff(p, i))}
DissectionsModDihedral(apply(v->isprime(v), [1..25]))
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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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