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A005809 Binomial coefficients C(3n,n).
(Formerly M2995)
45
1, 3, 15, 84, 495, 3003, 18564, 116280, 735471, 4686825, 30045015, 193536720, 1251677700, 8122425444, 52860229080, 344867425584, 2254848913647, 14771069086725, 96926348578605, 636983969321700, 4191844505805495, 27619435402363035 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 (including 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

In general, C(k*n,n)= C(k*n-1,n-1)*C((k*n)^2,2)/(3*n*C(k*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014

REFERENCES

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).

LINKS

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].

Paul Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), #13.5.1.

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

V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 436.

Random Variable, Ordinary generating function for binom(3n,n), Nov 2013.

W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595, 2013

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

FORMULA

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). - Philippe Deléham, 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

G.f.: 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

a(n) = (2n+1)*A001764(n). - Johannes W. Meijer, Aug 22 2013

a(n) = C(3*n-1,n-1)*C(9*n^2,2)/(3*n*C(3*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014

MAPLE

A005809:=n->binomial(3*n, n); seq(A005809(n), n=0..40); # Wesley Ivan Hurt, Mar 21 2014

MATHEMATICA

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*)

PROG

(Sage) [binomial(3*n, n) for n in xrange(0, 22)] # Zerinvary Lajos, 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

CROSSREFS

Cf. A007318.

Sequence in context: A115910 A106569 A026032 * A067122 A202336 A093593

Adjacent sequences:  A005806 A005807 A005808 * A005810 A005811 A005812

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 23 01:23 EST 2014. Contains 249836 sequences.