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A303731 Number of noncrossing path sets on n nodes up to rotation and reflection with each path having a prime number of nodes.

%I A303731 #9 May 03 2018 17:04:05
%S A303731 1,0,1,1,1,5,6,27,53,140,649,1297,6355,18038,63226,241741,744711,
%T A303731 3008107,10028056,37270169,138083464,488933323,1872525356,6763888465,
%U A303731 25498771059,95467533318,355595703773,1353873044078,5077809606803,19345857682140,73533468653115
%N A303731 Number of noncrossing path sets on n nodes up to rotation and reflection with each path having a prime number of nodes.
%H A303731 Andrew Howroyd, <a href="/A303731/b303731.txt">Table of n, a(n) for n = 0..500</a>
%o A303731 (PARI) \\ number of path sets with restricted path lengths
%o A303731 NCPathSetsModDihedral(v)={ my(n=#v);
%o A303731 my(p=serreverse(x/(1 + x*v[1] + sum(k=2, #v, (k*2^(k-3))*x^k*v[k])) + O(x^2*x^n) )/x);
%o A303731 my(vars=variables(p));
%o A303731 my(h=substvec(p + O(x^(n\2+1)),vars,apply(t->t^2, vars)));
%o A303731 my(q=x*deriv(p)/p);
%o A303731 my(R=v[1]*x + sum(i=1, (#v-1)\2, v[2*i+1]*2^(i-1)*x*(x^2*h)^i), Q=sum(i=1, #v\2, v[2*i]*2^(i-1)*(x^2*h)^i), T=intformal((p - 1 + sum(d=2,n, eulerphi(d)*substvec(q + O(x^(n\d+1)), vars, apply(t->t^d, vars))))/x));
%o A303731 O(x*x^n) + (1 + T + (1 + Q + (1+R)^2*h/(1-Q) + v[2]*x^2*h)/2)/2;
%o A303731 }
%o A303731 Vec(NCPathSetsModDihedral(vector(30, k, isprime(k))))
%Y A303731 Cf. A303729, A303732.
%K A303731 nonn
%O A303731 0,6
%A A303731 _Andrew Howroyd_, Apr 29 2018