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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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

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

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/(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

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

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)