0,2

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 (pbarry(AT)wit.ie), Jan 21 2007

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.

T. D. Noe, Table of n, a(n) for n=0..100

Milan Janjic, Two Enumerative Functions

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

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

N. T. Cameron, Random walks, trees, and extensions of Riordan group techniques

The g.f. R[ z_ ] below 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 (pbarry(AT)wit.ie), 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 (deutsch(AT)duke.poly.edu), 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 (lior.manor(AT)gmail.com) 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 (benoit7848c(AT)orange.fr), 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 (pbarry(AT)wit.ie), Jan 21 2007

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 )

Adjacent sequences: A005806 A005807 A005808 this_sequence A005810 A005811 A005812

Sequence in context: A115910 A106569 A026032 this_sequence A067122 A093593 A011900

nonn,easy,nice

njas