A332650 Number of polygonal cacti on 2n-1 unlabeled nodes with every polygon having an odd prime number of edges.

1, 1, 2, 4, 10, 30, 105, 400, 1654, 7229, 32944, 154749, 744973, 3655993, 18232812, 92162974, 471301437, 2434542190, 12687850499, 66646225443, 352548333438, 1876770716627, 10048289587337, 54079948967654, 292447643655469, 1588388448970674, 8661869330014601

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Wikipedia, Cactus graph

Index entries for sequences related to cacti

Example

a(3) = 2 because there are two cacti on 5 nodes which are a pentagon and 2 triangles joined at a node.

Prog

(PARI) \\ Here UCacti gives number of unrooted cacti with restricted polygons.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
RCacti(u)={my(v=[1]); while(#v<#u, my(g=x*Ser(v), g2=subst(g, x, x^2) + O(x^2*x^#v), r=sum(k=1, #u-1, my(c=u[k+1]); if(c, c*(g^k + g^(k%2)*g2^(k\2))))/2 + O(x^#u)); v=concat([1], EulerT(Vec(r, 1-serprec(r, x))))); v}
UCacti(u)={my(p=x*Ser(RCacti(u))); my(g(d)=subst(p + O(x*x^(#u\d)), x, x^d)); Vec(g(1) + sum(k=1, #u, my(c=u[k]); if(c, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/(2*k) - (g(1)^k)/2 + if(k%2==0, g(2)^(k/2) - g(1)^2*g(2)^(k/2-1))/4)))}
seq(n)={my(v=UCacti(vector(2*n-1, i, i>2 && isprime(i)))); vector(n, i, v[2*i-1])}

Crossrefs

Cf. A000083, A035085, A091487, A332649, A332651.

Sequence in context: A112846 A050397 A186021 * A091174 A005193 A317280

Adjacent sequences: A332647 A332648 A332649 * A332651 A332652 A332653

Keyword

nonn

Author

Andrew Howroyd, Feb 18 2020

Status

approved