0,2

Number of paths of length 3n in an n X n X n grid from (0,0,0) to (n,n,n).

Appears in Ramanujan's theory of elliptic functions of signature 3.

L. A. Aizenberg and A. P. Yuzhakov, "Integral representations and residues in multidimensional complex analysis", American Mathematical Society, 1983, p. 194.

G. E. Andrews, The well-poised thread ..., Ramanujan J., 1 (1997), 7-23; see Section 8.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 174.

M. Petkovsek et al., A=B, Peters, 1996, p. 22.

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

R. M. Dickau, 3-dimensional shortest-path diagrams

K. A. Penson and A. I. Solomon, Coherent states from combinatorial sequences.

B. Salvy, GFUN and the AGM.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ 1/2 * sqrt(3) * 27^n / (Pi*n) - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001

O.g.f.: hypergeom([1/3, 2/3], [1], 27*x); E.g.f.: hypergeom([1/3, 2/3], [1, 1], 27*x). Integral representation as n-th moment of a positive function on [0, 27]: a(n)= int( x^n* (-1/24*(3*sqrt(3)*hypergeom([2/3, 2/3], [4/3], 1/27*x)* GAMMA(2/3)^6*x^(1/3)-8*hypergeom([1/3, 1/3], [2/3], 1/27*x)*Pi^3)/Pi^3/x^(2/3)/GAMMA(2/3)^3), x=0..27), n=0, 1... . This representation is unique. - Karol PENSON (penson(AT)lptl.jussieu.fr), Nov 21, 2001

a(n)=sum(k=-n, +n, (-1)^k*binomial(2*n, n+k)^3) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 02 2005

a(n)=C(2n,n)*C(3n,n)=A104684(2n,n); - Paul Barry (pbarry(AT)wit.ie), Mar 14 2006

seq((3*n)!/(n!)^3, n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007

Sum [ (-1)^(k+n) Binomial[ 2n, k ]^3, {k, 0, 2n} ]

(PARI) a(n)=if(n<0, 0, (3*n)!/n!^3)

(PARI) a(n)=local(A, m); if(n<1, n==0, m=1; A=1+O(x); while(m<=n, m*=3; A=subst((1+2*x)*subst(A, x, (x/3)^3), x, serreverse(x*(1+x+x^2)/(1+2*x)^3/3+O(x^m)))); polcoeff(A, n))

Cf. A000984, A050983, A050984, A008977.

Adjacent sequences: A006477 A006478 A006479 this_sequence A006481 A006482 A006483

Sequence in context: A002432 A091800 A037959 this_sequence A002896 A004996 A001499

nonn,easy,nice

njas

More terms from Eric Weisstein (eric(AT)weisstein.com)