A035085 - OEIS

%I #22 Jan 18 2021 02:47:23

%S 1,1,0,1,1,2,2,5,7,16,28,63,131,301,673,1600,3773,9158,22319,55255,

%T 137563,345930,874736,2227371,5700069,14664077,37888336,98310195,

%U 256037795,669184336,1754609183,4614527680

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

%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.

%D F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.

%H Andrew Howroyd, <a href="/A035085/b035085.txt">Table of n, a(n) for n = 0..500</a>

%H F. Harary and R. Z. Norman, <a href="http://www.jstor.org/stable/1969824">The Dissimilarity Characteristic of Husimi Trees</a>, Annals of Mathematics, 58 1953, pp. 134-141.

%H F. Harary and G. E. Uhlenbeck, <a href="https://doi.org/10.1073/pnas.39.4.315">On the Number of Husimi Trees</a>, Proc. Nat. Acad. Sci. USA vol. 39 pp. 315-322 1953.

%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>

%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>

%F 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.

%o (PARI)

%o BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}

%o 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}

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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

%Y Cf. A035082, A035083, A035084.

%K nonn

%O 0,6

%A _Christian G. Bower_, Nov 15 1998

%E Terms a(32) and beyond from _Andrew Howroyd_, Aug 31 2018