A035085 Number of polygonal cacti (Husimi graphs) with n nodes.

1, 1, 0, 1, 1, 2, 2, 5, 7, 16, 28, 63, 131, 301, 673, 1600, 3773, 9158, 22319, 55255, 137563, 345930, 874736, 2227371, 5700069, 14664077, 37888336, 98310195, 256037795, 669184336, 1754609183, 4614527680

Offset: 0

Author

Christian G. Bower, Nov 15 1998

Keywords

• nonn

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
• F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• F. Harary and R. Z. Norman, The Dissimilarity Characteristic of Husimi Trees, Annals of Mathematics, 58 1953, pp. 134-141.
• F. Harary and G. E. Uhlenbeck, On the Number of Husimi Trees, Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953.
• Index entries for sequences related to cacti
• Index entries for sequences related to trees

Crossrefs

Cf. A035082, A035083, A035084.

Programs

PARI

BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
DIK(p,n)={(sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d))) + ((1+p)^2/(1-subst(p, x, x^2))-1)/2)/2}
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*Ser(EulerT(Vec(BIK(p)-1)-Vec(p)))); Vec(1 + DIK(p, n) - (p^2 + subst(p, x, x^2))/2 - p*(BIK(p)-1-p))} \\ Andrew Howroyd, Aug 31 2018

Formula

G.f.: A(x) = B(x) + C(x) - B(x)*D(x) where B, C, D are g.f.s of A035082, A035083, A035084, respectively.

Extensions

Terms a(32) and beyond from Andrew Howroyd, Aug 31 2018