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%I #10 Dec 23 2020 01:51:32
%S 1,1,2,4,10,30,105,400,1654,7229,32944,154749,744973,3655993,18232812,
%T 92162974,471301437,2434542190,12687850499,66646225443,352548333438,
%U 1876770716627,10048289587337,54079948967654,292447643655469,1588388448970674,8661869330014601
%N Number of polygonal cacti on 2n-1 unlabeled nodes with every polygon having an odd prime number of edges.
%H Andrew Howroyd, <a href="/A332650/b332650.txt">Table of n, a(n) for n = 1..200</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cactus_graph">Cactus graph</a>
%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%e a(3) = 2 because there are two cacti on 5 nodes which are a pentagon and 2 triangles joined at a node.
%o (PARI) \\ Here UCacti gives number of unrooted cacti with restricted polygons.
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o 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}
%o 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)))}
%o seq(n)={my(v=UCacti(vector(2*n-1, i, i>2 && isprime(i)))); vector(n, i, v[2*i-1])}
%Y Cf. A000083, A035085, A091487, A332649, A332651.
%K nonn
%O 1,3
%A _Andrew Howroyd_, Feb 18 2020
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