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A005809
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Binomial coefficients C(3n,n).
(Formerly M2995)
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40
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1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, 30045015, 193536720, 1251677700, 8122425444, 52860229080, 344867425584, 2254848913647, 14771069086725, 96926348578605, 636983969321700, 4191844505805495, 27619435402363035
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OFFSET
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0,2
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COMMENTS
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Number of paths in Z x Z starting at (0,0) and ending at (3n,0) using steps in {(1,1),(1,-2)}.
Number of even trees with 2n edges and one distinguished vertex. Even trees are rooted plane trees where every vertex (includig root) has even degree.
Hankel transform is 3^n*A051255(n), where A051255 is the Hankel transform of C(3n,n)/(2n+1). - Paul Barry, Jan 21 2007
T(n,k) is the number of stack polyominoes inscribed in an (n+1)x(n+1) box. Equivalently, T(n,k) is the number of unimodal compositions with n+1 parts in which the maximum value of the parts is n+1. For instance, for n=2, we have the following compositions: (3,3,3), (2,3,3), (1,3,3), (3,3,1), (3,3,2), (2,2,3), (1,2,3), (2,3,1), (1,1,3), (1,3,1), (3,1,1), (2,3,2), (1,3,2), (3,2,1), (3,2,2). [Emanuele Munarini, Apr 07 2011]
Conjecture: a(n)==3 (mod n^3) iff n is an odd prime. - Gary Detlefs, Mar 23 2013
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques
T. C. Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers
Milan Janjic, Two Enumerative Functions
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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FORMULA
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The g.f. R[ z_ ] below (in the Mathematica field) was found by Kurt Persson (kurt(AT)math.chalmers.se) and communicated by Einar Steingrimsson (einar(AT)math.chalmers.se).
Using Stirling's formula in A000142 it easy to get the asymptotic expression a(n) ~ 1/2 * (27/4)^n / sqrt(Pi*n / 3) - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
a(n)=sum{k=0..n, C(n, k)C(2n, k) } - Paul Barry, May 15 2003
G.f. = 1/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e. (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z); - Emeric Deutsch, May 22 2003
a(n) ~ 1/2*3^(1/2)*pi^(-1/2)*n^(-1/2)*2^(-2*n)*3^(3*n)*{1 - 7/72*n^-1 + 49/10368*n^-2 + 6425/2239488*n^-3 - ...} - Joe Keane (jgk(AT)jgk.org), Nov 07 2003
a(n) = A006480(n)/A000984(n) - Lior Manor May 04 2004
a(n)=sum_{0<=i_1<=n, 0<= i_2<=n}binomial(n, i_1)*binomial(n, i_2)*binomial(n, i_1+i_2) - Benoit Cloitre, Oct 14 2004
a(n)=sum{k=0..n, A109971(k)*3^k}; a(0)=1, a(n)=sum{k=0..n, 3^k*C(3n-k,n-k)2k/(3n-k)}, n>0; - Paul Barry, Jan 21 2007
a(n)=A085478(2n,n). [From Philippe DELEHAM, Sep 17 2009]
E.g.f: F(1/3,2/3;1/2,1;27*x/4), where F(a1,a2;b1,b2;z) is a hypergeometric series. [Emanuele Munarini, Apr 12 2011]
a(n) = sum(k=0..n, binomial(2*n+k-1,k)). [Arkadiusz Wesolowski, Apr 02 2012]
The G.F. is cos{(1/3)asin[(27x/4)^(1/2)]}/(1-27x/4)^(1/2). [Tom Copeland, May 24 2012]
G.f.: A(x) = 1 + 6*x/(G(0)-6*x) where G(k)= (2*k+2)*(2*k+1) + 3*x*(3*k+1)*(3*k+2) - 6*x*(k+1)*(2*k+1)*(3*k+4)*(3*k+5)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jun 30 2012
2*n*(2*n-1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Feb 05 2013
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MATHEMATICA
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R[ z_ ] := ((2-18*z + 27*z^2 + 3^(3/2)*z^(3/2)*(27*z-4)^(1/2))/2)^(1/3); f[ z_ ] := ( (R[ z ])^3 + (1-3*z)*(R[ z ])^2 + (1-6*z)*R[ z ] )/( (R[ z ])^4 + (1-6*z)*(R[ z ])^2 + (6*z-1)^2 )
Table[Binomial[3*n, n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2011*)
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PROG
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(Sage) [binomial(3*n, n) for n in xrange(0, 22)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 16 2009]
(Maxima) makelist(binomial(3*n, n), n, 0, 100); [Emanuele Munarini, Apr 07 2011]
(MAGMA) [ Binomial(3*n, n): n in [0..150] ]; // Vincenzo Librandi, Apr 21 2011
(Haskell)
a005809 n = a007318 (3*n) n -- Reinhard Zumkeller, May 06 2012
(PARI) a(n)=binomial(3*n, n) \\ Charles R Greathouse IV, Nov 20 2012
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CROSSREFS
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Cf. A007318.
Sequence in context: A115910 A106569 A026032 * A067122 A202336 A093593
Adjacent sequences: A005806 A005807 A005808 * A005810 A005811 A005812
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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