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 A332650 Number of polygonal cacti on 2n-1 unlabeled nodes with every polygon having an odd prime number of edges. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 Wikipedia, Cactus graph 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

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Last modified April 15 03:00 EDT 2021. Contains 342974 sequences. (Running on oeis4.)