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

1, 0, 1, 1, 1, 5, 6, 27, 53, 140, 649, 1297, 6355, 18038, 63226, 241741, 744711, 3008107, 10028056, 37270169, 138083464, 488933323, 1872525356, 6763888465, 25498771059, 95467533318, 355595703773, 1353873044078, 5077809606803, 19345857682140, 73533468653115

Offset

0,6

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Prog

(PARI) \\ number of path sets with restricted path lengths
NCPathSetsModDihedral(v)={ my(n=#v);
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);
my(vars=variables(p));
my(h=substvec(p + O(x^(n\2+1)), vars, apply(t->t^2, vars)));
my(q=x*deriv(p)/p);
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(x*x^n) + (1 + T + (1 + Q + (1+R)^2*h/(1-Q) + v[2]*x^2*h)/2)/2;
}
Vec(NCPathSetsModDihedral(vector(30, k, isprime(k))))

Crossrefs

Cf. A303729, A303732.

Sequence in context: A022163 A048060 A320665 * A298175 A298144 A115761

Adjacent sequences: A303728 A303729 A303730 * A303732 A303733 A303734

Keyword

nonn

Author

Andrew Howroyd, Apr 29 2018

Status

approved