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A006632 a(n) = 3*binomial(4*n-1, n-1)/(4*n-1).

%I A006632 M2997 #113 Sep 01 2025 03:06:41
%S A006632 1,3,15,91,612,4389,32890,254475,2017356,16301164,133767543,
%T A006632 1111731933,9338434700,79155435870,676196049060,5815796869995,
%U A006632 50318860986108,437662920058980,3824609516638444,33563127932394060,295655735395397520,2613391671568320765
%N A006632 a(n) = 3*binomial(4*n-1, n-1)/(4*n-1).
%C A006632 a(n) is the number of ordered trees (A000108) with 3n-1 edges in which every non-leaf vertex has exactly two leaf children (no restriction on non-leaf children). For example, a(2) counts the 3 trees
%C A006632   \/......\/......\/
%C A006632   .\|/...\|/....\|/  . - _David Callan_, Aug 22 2014
%C A006632 a(n) is the number of lattice paths from (0,0) to (3n,n) using only the steps (1,0) and (0,1) and which are strictly below the line y = x/3 except at the path's endpoints. - _Lucas A. Brown_, Aug 21 2020
%C A006632 This is instance k = 3 of the family {c(k, n)}_{n>=1} given in a comment in A130564. - _Wolfdieter Lang_, Feb 04 2024
%D A006632 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 A006632 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006632 G. C. Greubel, <a href="/A006632/b006632.txt">Table of n, a(n) for n = 1..1000</a>
%H A006632 O. Aichholzer, A. Asinowski and T. Miltzow, <a href="http://arxiv.org/abs/1403.5546">Disjoint compatibility graph of non-crossing matchings of points in convex position</a>, arXiv preprint arXiv:1403.5546 [math.CO], 2014.
%H A006632 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A006632 Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%H A006632 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=438">Encyclopedia of Combinatorial Structures 438</a>
%H A006632 Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H A006632 J. Sawada, J. Sears, A. Trautrim, and A. Williams, <a href="https://arxiv.org/abs/2308.12405">Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees</a>, arXiv:2308.12405 [math.CO], 2023.
%F A006632 a(n) = binomial(4*n-1, n)/(4*n-1) = 3*binomial(4*n-2, n-1) - binomial(4*n-2, n). - _David Callan_, Sep 15 2004
%F A006632 G.f.: g^3 where g = 1+x*g^4 is the g.f. of A002293. - _Mark van Hoeij_, Nov 11 2011
%F A006632 a(n) = (3/4)*binomial(4*n,n)/(4*n-1). - _Bruno Berselli_, Jan 17 2014
%F A006632 From _Wolfdieter Lang_, Feb 06 2020: (Start)
%F A006632 G.f.: (3/4)*(1 - hypergeometric3F2([-1, 1, 2]/4, [1, 2]/3, (4^4/3^3)*x)).
%F A006632 E.g.f.: (3/4)*(1 - hypergeometric3F3([-1, 1, 2]/4, [1, 2, 3]/3, (4^4/3^3)*x)). (End)
%F A006632 D-finite with recurrence 3*n*(3*n-1)*(3*n-2)*a(n) -8*(4*n-5)*(4*n-3)*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, May 07 2021
%F A006632 a(n) = (2n-1)*A000260(n). - _F. Chapoton_, Jul 15 2021
%F A006632 G.f. A(x) satisfies: A(x) = x / (1 - A(x))^3. - _Ilya Gutkovskiy_, Nov 03 2021
%F A006632 G.f.: x*( Sum_{n >= 0} binomial(4*n+3, n)*x^n ) / ( Sum_{n >= 0} binomial(4*n, n)*x^n ) = x*( Sum_{n >= 0} binomial(4*n+3, n)*x^n ) / ( 1 + 4*x*Sum_{n >= 0} binomial(4*n+3, n)*x^n ). - _Peter Bala_, Dec 13 2024
%F A006632 Working with a offset of 0, the g.f. A(x) = 1 + 3*x + 15*x^2 + ... is uniquely determined by the conditions A(0) = 1 and [x^n] A(x)^(-n) = -3 for all n >= 1. - _Peter Bala_, Jul 24 2025
%p A006632 A006632:=n->3*binomial(4*n-1,n-1)/(4*n-1): seq(A006632(n), n=1..30); # _Wesley Ivan Hurt_, Oct 23 2017
%t A006632 InverseSeries[Series[y*(1-y)^3, {y, 0, 24}], x] (* then A(x)=y(x) *) (* _Len Smiley_, Apr 07 2000 *)
%t A006632 a[ n_] := If[n<1, 0, Binomial[4 n - 2, n - 1] / n]; (* _Michael Somos_, Aug 22 2014 *)
%o A006632 (PARI) a(n) = 3*binomial(4*n-1, n-1)/(4*n-1) \\ _Felix Fröhlich_, Oct 23 2017
%o A006632 (Magma)
%o A006632 A006632:= func< n | Binomial(4*n-2,n-1)/n >;
%o A006632 [A006632(n): n in [1..40]]; // _G. C. Greubel_, Sep 01 2025
%o A006632 (SageMath)
%o A006632 def A006632(n): return binomial(4*n-2,n-1)//n
%o A006632 print([A006632(n) for n in range(1,41)]) # _G. C. Greubel_, Sep 01 2025
%Y A006632 Cf. A000108, A000260, A002293, A006013, A069271, (1/2)*A112385, A120588, A130564.
%K A006632 nonn,easy,changed
%O A006632 1,2
%A A006632 _Simon Plouffe_