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%I M4214 #63 Feb 23 2024 10:16:12
%S 1,6,33,182,1020,5814,33649,197340,1170585,7012200,42364476,257854776,
%T 1579730984,9734161206,60290077905,375138262520,2343880406595,
%U 14699630061270,92502956574105,583920410197950,3696470074992240,23461536762704040,149270218961671548
%N From generalized Catalan numbers.
%C It appears that this is the self-convolution of A001764 starting 1, 3, 12, ... . - _Alon Regev_, Aug 07 2015
%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="/A006630/b006630.txt">Table of n, a(n) for n = 0..200</a>
%H Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, <a href="https://arxiv.org/abs/2303.10986">Refined product formulas for Tamari intervals</a>, arXiv:2303.10986 [math.CO], 2023.
%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.
%F G.f.: 3_F_2 ( [ 2, 8/3, 7/3 ]; [ 4, 7/2 ]; 27 x / 4 ).
%F a(n) = C(3n+6, n)*2/(n+2). - _Henry Bottomley_, Sep 24 2001
%F G.f.: (1-RootOf(x-t*(1-t)^2,t))^(-6) (algebraic function in Maple notation). - _Mark van Hoeij_, Nov 08 2011
%F G.f.: ((1/sqrt((3/4)*x)*sin((1/3)*asin(sqrt((27/4)*x)))-1)/x)^2. - _Vladimir Kruchinin_, Oct 03 2022
%F a(n) = (n+1)/2 * A000139(n+2). - _F. Chapoton_, Feb 23 2024
%t Table[Binomial[3 n + 6, n] 2 / (n + 2), {n, 0, 25}] (* _Vincenzo Librandi_, Aug 07 2015 *)
%t CoefficientList[Series[(-1 + (2*Sin[(1/3)*ArcSin[(3*Sqrt[3]*Sqrt[x])/2]]) / (Sqrt[3]*Sqrt[x]))^2/x^2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 03 2022, after _Vladimir Kruchinin_ *)
%o (Magma) [Binomial(3*n+6, n)*2/(n+2): n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2015
%o (PARI) a(n) = binomial(3*n+6, n)*2/(n+2); \\ _Andrew Howroyd_, Nov 06 2017
%o (Maxima) taylor(((1/sqrt(3/4*x)*sin(1/3*asin(sqrt(27/4*x)))-1)/x)^2,x,0,17); /* _Vladimir Kruchinin_, Oct 03 2022 */
%o (Maxima) makelist(binomial(3*n+6, n)*2/(n+2),n,0,30); /* _Vladimir Kruchinin_, Oct 03 2022 */
%Y Column 3 of A092276.
%Y Closely related to A000139.
%K nonn,easy
%O 0,2
%A _Simon Plouffe_
%E More terms from Christopher Lund (clund(AT)san.rr.com), Apr 16 2002
%E a(21)-a(22) from _Vincenzo Librandi_, Aug 07 2015
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