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A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.

%I A006629 M3542 #61 Aug 30 2025 09:14:29
%S A006629 1,4,18,88,455,2448,13566,76912,444015,2601300,15426840,92431584,
%T A006629 558685348,3402497504,20858916870,128618832864,797168807855,
%U A006629 4963511449260,31032552351570,194743066471800,1226232861415695
%N A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.
%C A006629 Sum of root degrees of all noncrossing trees on nodes on a circle. - _Emeric Deutsch_
%D A006629 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 A006629 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006629 Andrew Howroyd, <a href="/A006629/b006629.txt">Table of n, a(n) for n = 0..200</a>
%H A006629 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 A006629 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 A006629 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 A006629 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A006629 a(n) = 2*binomial(3*n+3,n)/(n+2). - _Emeric Deutsch_
%F A006629 a(n) = (n+1) * A000139(n+1). - _F. Chapoton_, Feb 23 2024
%F A006629 G.f.: hypergeom( [ 2, 5/3, 4/3 ]; [ 3, 5/2 ]; 27*x/4 ).
%F A006629 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 A006629 G.f.: (((4*sin(arcsin((3*sqrt(3*x))/2)/3))/(sqrt(3*x))-1)^2-1)/(4*x). - _Vladimir Kruchinin_, Feb 17 2023
%F A006629 E.g.f.: hypergeom([4/3, 5/3, 2]; [1, 5/2, 3]; 27*x/4). - _G. C. Greubel_, Aug 29 2025
%t A006629 Table[2*Binomial[3*n+3,n]/(n+2), {n,0,40}] (* _G. C. Greubel_, Aug 29 2025 *)
%o A006629 (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
%o A006629 (Magma)
%o A006629 A006629:= func< n | 2*Binomial(3*n+3,n)/(n+2) >;
%o A006629 [A006629(n): n in [0..40]]; // _G. C. Greubel_, Aug 29 2025
%o A006629 (SageMath)
%o A006629 def A006629(n): return 2*binomial(3*n+3,n)//(n+2)
%o A006629 print([A006629(n) for n in range(41)]) # _G. C. Greubel_, Aug 29 2025
%Y A006629 Column 2 of A092276.
%Y A006629 Cf. A000139, A001764, A006013, A006630, A006631, A230547, A233657.
%K A006629 nonn,easy,changed
%O A006629 0,2
%A A006629 _Simon Plouffe_, _N. J. A. Sloane_
%E A006629 More precise definition from _Paul D. Hanna_, May 10 2008