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A006480
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De Bruijn's S(3,n): (3n)!/(n!)^3.
(Formerly M4284)
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25
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1, 6, 90, 1680, 34650, 756756, 17153136, 399072960, 9465511770, 227873431500, 5550996791340, 136526995463040, 3384731762521200, 84478098072866400, 2120572665910728000, 53494979785374631680, 1355345464406015082330
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
S(s,n) = sum_{k=0..2n} (-1)^(k+n) * binomial(2n, k)^s. The formula S(3,n) = (3n)!/(n!)^3 is due to Dixon (according to W. N. Bailey 1935). [Charles R Greathouse IV, Dec 28 2011]
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REFERENCES
| 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.
H. J. Brothers, Pascal's Prism: Supplementary Material, http://www.brotherstechnology.com/docs/Pascal's_Prism_(supplement).pdf.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 174.
N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjav´k, Iceland DMTCS proc. AO, 2011, 599-610; http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/download/dmAO0153/3610.
M. Petkovsek et al., A=B, Peters, 1996, p. 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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, Binomial Sums
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FORMULA
| 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
G.f. satisfies: A(x^3) = A( x*(1+3*x+9*x^2)/(1+6*x)^3 )/(1+6*x). [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 29 2010]
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MAPLE
| seq((3*n)!/(n!)^3, n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 28 2007
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MATHEMATICA
| Sum [ (-1)^(k+n) Binomial[ 2n, k ]^3, {k, 0, 2n} ]
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PROG
| (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))
(MAGMA) [Factorial(3*n)/(Factorial(n))^3: n in [0..20] ]; // Vincenzo Librandi, Aug 20 2011
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CROSSREFS
| Cf. A000984, A050983, A050984, A008977.
Cf. A181545. [From Paul D. Hanna (pauldhanna(AT)juno.com), Oct 29 2010]
Sequence in context: A002432 A091800 A037959 * A138462 A002896 A004996
Adjacent sequences: A006477 A006478 A006479 * A006481 A006482 A006483
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Eric Weisstein (eric(AT)weisstein.com)
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