

A035085


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


4



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
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OFFSET

0,6


REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and TreeLike Structures, Camb. 1998, p. 301.
F. Harary and E. M. Palmer, Graphical Enumeration, p. 71
F. Harary and R. Z. Norman "The Dissimilarity Characteristic of Husimi Trees" Annals of Mathematics, 58 1953, pp. 134141
F. Harary and G. E. Uhlenbeck "On the Number of Husimi Trees" Proc. Nat. Acad. Sci. USA vol. 39 pp. 315322 1953


LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..500
Index entries for sequences related to cacti
Index entries for sequences related to trees


FORMULA

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


PROG

(PARI)
BIK(p)={(1/(1p) + (1+p)/subst(1p, 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/(1subst(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)1p))} \\ Andrew Howroyd, Aug 31 2018


CROSSREFS

Cf. A035082, A035083, A035084.
Sequence in context: A047083 A238422 A327019 * A208238 A127413 A145344
Adjacent sequences: A035082 A035083 A035084 * A035086 A035087 A035088


KEYWORD

nonn


AUTHOR

Christian G. Bower, Nov 15 1998


EXTENSIONS

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


STATUS

approved



