|
%I M1152 #38 Feb 26 2020 09:57:28
%S 1,1,1,2,4,8,19,48,126,355,1037,3124,9676,30604,98473,321572,1063146,
%T 3552563,11982142,40746208,139573646,481232759,1669024720,5819537836,
%U 20390462732,71762924354,253601229046,899586777908,3202234779826,11435967528286,40964243249727
%N Number of triangular cacti with 2n+1 nodes (n triangles).
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 306, (4.2.35).
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 73, (3.4.21).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A003081/b003081.txt">Table of n, a(n) for n = 0..1000</a>
%H Maryam Bahrani and Jérémie Lumbroso, <a href="http://arxiv.org/abs/1608.01465">Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition</a>, arXiv:1608.01465 [math.CO], 2016.
%H P. Leroux and B. Miloudi, <a href="http://www.labmath.uqam.ca/~annales/volumes/16-1/PDF/053-080.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992) pp. 53-80.
%H P. Leroux and B. Miloudi, <a href="/A000081/a000081_2.pdf">Généralisations de la formule d'Otter</a>, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>
%F a(n)=b(2n+1). A003080(n)=c(2n+1).
%F G.f.: B(x)=C(x)+(C(x^3)-C(x)^3)/3.
%F G.f.: g(x) + x*(g(x^3) - g(x)^3)/3 where g(x) is the g.f. of A003080. - _Andrew Howroyd_, Feb 18 2020
%t terms = 31;
%t nmax = 2 terms;
%t A[_] = 0;
%t Do[A[x_] = x Exp[Sum[(A[x^n]^2 + A[x^(2n)])/(2n), {n, 1, terms}]] + O[x]^nmax // Normal, {nmax}];
%t g[x_] = (A[x] /. x^k_ -> x^((k - 1)/2)) - x + 1;
%t g[x] + x((g[x^3] - g[x]^3)/3) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2020, after _Andrew Howroyd_ *)
%Y Column k=3 of A332649.
%Y Cf. A003080, A034940, A034941.
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_
%E Extended with formula by _Christian G. Bower_, 10/98
|
