cp's OEIS Frontend

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

%I A035085 #22 Jan 18 2021 02:47:23
%S A035085 1,1,0,1,1,2,2,5,7,16,28,63,131,301,673,1600,3773,9158,22319,55255,
%T A035085 137563,345930,874736,2227371,5700069,14664077,37888336,98310195,
%U A035085 256037795,669184336,1754609183,4614527680
%N A035085 Number of polygonal cacti (Husimi graphs) with n nodes.
%D A035085 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 301.
%D A035085 F. Harary and E. M. Palmer, Graphical Enumeration, p. 71.
%H A035085 Andrew Howroyd, <a href="/A035085/b035085.txt">Table of n, a(n) for n = 0..500</a>
%H A035085 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 A035085 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 A035085 <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%H A035085 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A035085 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 A035085 (PARI)
%o A035085 BIK(p)={(1/(1-p) + (1+p)/subst(1-p, x, x^2))/2}
%o A035085 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 A035085 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A035085 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 A035085 Cf. A035082, A035083, A035084.
%K A035085 nonn
%O A035085 0,6
%A A035085 _Christian G. Bower_, Nov 15 1998
%E A035085 Terms a(32) and beyond from _Andrew Howroyd_, Aug 31 2018