%I M3542 #51 Feb 23 2024 08:00:06
%S 1,4,18,88,455,2448,13566,76912,444015,2601300,15426840,92431584,
%T 558685348,3402497504,20858916870,128618832864,797168807855,
%U 4963511449260,31032552351570,194743066471800,1226232861415695
%N Self-convolution 4th power of A001764, which enumerates ternary trees.
%C Sum of root degrees of all noncrossing trees on nodes on a circle. - _Emeric Deutsch_
%D H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A006629/b006629.txt">Table of n, a(n) for n = 0..200</a>
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%H Joris Nieuwveld, <a href="https://arxiv.org/abs/2108.11382">Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding</a>, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
%H C. H. Pah, <a href="http://dx.doi.org/10.1007/s10955-010-9989-5">Single polygon counting on Cayley Tree of order 3</a>, J. Stat. Phys. 140 (2010) 198-207.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) = 2*binomial(3*n+3,n)/(n+2). - _Emeric Deutsch_
%F a(n) = (n+1) * A000139(n+1). - _F. Chapoton_, Feb 23 2024
%F G.f.: 3_F_2 ( [ 2, 5/3, 4/3 ]; [ 3, 5/2 ]; 27 x / 4 ).
%F G.f.: A(x) = G(x)^4 where G(x) = 1 + x*G(x)^3 = g.f. of A001764 giving a(n)=C(3n+m-1,n)*m/(2n+m) at power m=4 with offset n=0. - _Paul D. Hanna_, May 10 2008
%F G.f.: (((4*sin(arcsin((3*sqrt(3*x))/2)/3))/(sqrt(3*x))-1)^2-1)/(4*x). - _Vladimir Kruchinin_, Feb 17 2023
%o (PARI) a(n)=my(m=4);binomial(3*n+m-1,n)*m/(2*n+m) /* 4th power of A001764 with offset n=0 */ \\ _Paul D. Hanna_, May 10 2008
%Y Column 2 of A092276.
%Y Cf. A000139.
%K nonn,easy
%O 0,2
%A _Simon Plouffe_, _N. J. A. Sloane_
%E More precise definition from _Paul D. Hanna_, May 10 2008
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