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A006632 a(n) = 3*binomial(4*n-1,n-1)/(4*n-1).
(Formerly M2997)
25

%I M2997

%S 1,3,15,91,612,4389,32890,254475,2017356,16301164,133767543,

%T 1111731933,9338434700,79155435870,676196049060,5815796869995,

%U 50318860986108,437662920058980,3824609516638444,33563127932394060,295655735395397520,2613391671568320765

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

%C 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 \/......\/......\/

%C .\|/...\|/....\|/ . - _David Callan_, Aug 22 2014

%C 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

%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 O. Aichholzer, A. Asinowski, 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 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 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=438">Encyclopedia of Combinatorial Structures 438</a>

%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.

%F 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 G.f.: g^3 where g = 1+x*g^4 is the g.f. of A002293. - _Mark van Hoeij_, Nov 11 2011

%F a(n) = (3/4)*binomial(4*n,n)/(4*n-1). - _Bruno Berselli_, Jan 17 2014

%F From _Wolfdieter Lang_, Feb 06 2020:(Start)

%F G.f.: (3/4)*(1 - hypergeom([-1, 1, 2]/4, [1, 2]/3, (4^4/3^3)*x)),

%F E.g.f.: (3/4)*(1 - hypergeom([-1, 1, 2]/4, [1, 2, 3]/3, (4^4/3^3)*x)). (End)

%p 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 InverseSeries[Series[y*(1-y)^3, {y, 0, 24}], x] (* then A(x)=y(x) *) (* _Len Smiley_, Apr 07 2000 *)

%t a[ n_] := If[ n < 1, 0, Binomial[4 n - 2, n - 1] / n]; (* _Michael Somos_, Aug 22 2014 *)

%o (PARI) a(n) = 3*binomial(4*n-1, n-1)/(4*n-1) \\ _Felix Fröhlich_, Oct 23 2017

%Y A112385 divided by 2.

%Y Cf. A000108, A002293, A006013, A120588.

%K nonn,easy

%O 1,2

%A _Simon Plouffe_

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Last modified April 10 11:15 EDT 2021. Contains 342845 sequences. (Running on oeis4.)