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 A006632 a(n) = 3*binomial(4*n-1,n-1)/(4*n-1). (Formerly M2997) 29
 1, 3, 15, 91, 612, 4389, 32890, 254475, 2017356, 16301164, 133767543, 1111731933, 9338434700, 79155435870, 676196049060, 5815796869995, 50318860986108, 437662920058980, 3824609516638444, 33563127932394060, 295655735395397520, 2613391671568320765 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 \/......\/......\/ .\|/...\|/....\|/ . - David Callan, Aug 22 2014 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 REFERENCES 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. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Table of n, a(n) for n=1..22. O. Aichholzer, A. Asinowski and T. Miltzow, Disjoint compatibility graph of non-crossing matchings of points in convex position, arXiv preprint arXiv:1403.5546 [math.CO], 2014. Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020. Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 438 Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023. FORMULA 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 G.f.: g^3 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011 a(n) = (3/4)*binomial(4*n,n)/(4*n-1). - Bruno Berselli, Jan 17 2014 From Wolfdieter Lang, Feb 06 2020: (Start) G.f.: (3/4)*(1 - hypergeom([-1, 1, 2]/4, [1, 2]/3, (4^4/3^3)*x)). E.g.f.: (3/4)*(1 - hypergeom([-1, 1, 2]/4, [1, 2, 3]/3, (4^4/3^3)*x)). (End) 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 a(n) = (2n-1)*A000260(n). - F. Chapoton, Jul 15 2021 G.f. A(x) satisfies: A(x) = x / (1 - A(x))^3. - Ilya Gutkovskiy, Nov 03 2021 MAPLE A006632:=n->3*binomial(4*n-1, n-1)/(4*n-1): seq(A006632(n), n=1..30); # Wesley Ivan Hurt, Oct 23 2017 MATHEMATICA InverseSeries[Series[y*(1-y)^3, {y, 0, 24}], x] (* then A(x)=y(x) *) (* Len Smiley, Apr 07 2000 *) a[ n_] := If[ n < 1, 0, Binomial[4 n - 2, n - 1] / n]; (* Michael Somos, Aug 22 2014 *) PROG (PARI) a(n) = 3*binomial(4*n-1, n-1)/(4*n-1) \\ Felix Fröhlich, Oct 23 2017 CROSSREFS A112385 divided by 2. Cf. A000108, A000260, A002293, A006013, A069271, A120588. Sequence in context: A364740 A366090 A171790 * A366056 A159928 A020018 Adjacent sequences: A006629 A006630 A006631 * A006633 A006634 A006635 KEYWORD nonn,easy AUTHOR Simon Plouffe STATUS approved

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Last modified September 29 06:43 EDT 2023. Contains 365757 sequences. (Running on oeis4.)