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 A295419 Number of dissections of an n-gon by nonintersecting diagonals into polygons with a prime number of sides counted up to rotations and reflections. 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,3 COMMENTS 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. 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 DissectionsModDihedral[v_] := Module[{n = Length[v], q, vars, u, R, Q, T, p}, q = Table[0, {n}]; q[] = 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[]]; vars = Variables[q[]]; 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]]; DissectionsModDihedral[Boole[PrimeQ[#]]& /@ Range[1, 31]] (* Jean-François Alcover, Sep 25 2019, after Andrew Howroyd *) PROG (PARI) \\ number of dissections into parts defined by set. DissectionsModDihedral(v)={my(n=#v); my(q=vector(n)); q=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); my(vars=variables(q)); 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])) CROSSREFS Cf. A001004, A290571, A290646, A290722, A290816, A295260, A295634. Sequence in context: A034906 A018323 A151380 * A290816 A181070 A226659 Adjacent sequences:  A295416 A295417 A295418 * A295420 A295421 A295422 KEYWORD nonn AUTHOR Andrew Howroyd, Nov 22 2017 STATUS approved

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Last modified November 19 22:34 EST 2019. Contains 329323 sequences. (Running on oeis4.)