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 A006630 From generalized Catalan numbers. (Formerly M4214) 7
 1, 6, 33, 182, 1020, 5814, 33649, 197340, 1170585, 7012200, 42364476, 257854776, 1579730984, 9734161206, 60290077905, 375138262520, 2343880406595, 14699630061270, 92502956574105, 583920410197950, 3696470074992240, 23461536762704040, 149270218961671548 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS It appears that this is the self-convolution of A001764 starting 1, 3, 12, ... . - Alon Regev, Aug 07 2015 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 Andrew Howroyd, Table of n, a(n) for n = 0..200 Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023. Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2. FORMULA G.f.: 3_F_2 ( [ 2, 8/3, 7/3 ]; [ 4, 7/2 ]; 27 x / 4 ). a(n) = C(3n+6, n)*2/(n+2). - Henry Bottomley, Sep 24 2001 G.f.: (1-RootOf(x-t*(1-t)^2,t))^(-6) (algebraic function in Maple notation). - Mark van Hoeij, Nov 08 2011 G.f.: ((1/sqrt((3/4)*x)*sin((1/3)*asin(sqrt((27/4)*x)))-1)/x)^2. - Vladimir Kruchinin, Oct 03 2022 a(n) = (n+1)/2 * A000139(n+2). - F. Chapoton, Feb 23 2024 MATHEMATICA Table[Binomial[3 n + 6, n] 2 / (n + 2), {n, 0, 25}] (* Vincenzo Librandi, Aug 07 2015 *) 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 *) PROG (Magma) [Binomial(3*n+6, n)*2/(n+2): n in [0..25]]; // Vincenzo Librandi, Aug 07 2015 (PARI) a(n) = binomial(3*n+6, n)*2/(n+2); \\ Andrew Howroyd, Nov 06 2017 (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 */ (Maxima) makelist(binomial(3*n+6, n)*2/(n+2), n, 0, 30); /* Vladimir Kruchinin, Oct 03 2022 */ CROSSREFS Column 3 of A092276. Closely related to A000139. Sequence in context: A009162 A012718 A297221 * A367850 A180035 A360717 Adjacent sequences: A006627 A006628 A006629 * A006631 A006632 A006633 KEYWORD nonn,easy AUTHOR Simon Plouffe EXTENSIONS More terms from Christopher Lund (clund(AT)san.rr.com), Apr 16 2002 a(21)-a(22) from Vincenzo Librandi, Aug 07 2015 STATUS approved

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Last modified August 8 03:39 EDT 2024. Contains 375018 sequences. (Running on oeis4.)