Search: id:A006480
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%I A006480 M4284
%S A006480 1,6,90,1680,34650,756756,17153136,399072960,9465511770,227873431500,
%T A006480 5550996791340,136526995463040,3384731762521200,84478098072866400,
%U A006480 2120572665910728000,53494979785374631680,1355345464406015082330
%N A006480 De Bruijn's s(3,n): (3n)!/(n!)^3.
%C A006480 Number of paths of length 3n in an n X n X n grid from (0,0,0) to (n,
               n,n).
%C A006480 Appears in Ramanujan's theory of elliptic functions of signature 3.
%D A006480 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006480 L. A. Aizenberg and A. P. Yuzhakov, "Integral representations and residues 
               in multidimensional complex analysis", American Mathematical Society, 
               1983, p. 194.
%D A006480 G. E. Andrews, The well-poised thread ..., Ramanujan J., 1 (1997), 7-23; 
               see Section 8.
%D A006480 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 174.
%D A006480 M. Petkovsek et al., A=B, Peters, 1996, p. 22.
%H A006480 T. D. Noe, <a href="b006480.txt">Table of n, a(n) for n=0..100</a>
%H A006480 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/path3d.html">
               3-dimensional shortest-path diagrams</a>
%H A006480 K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/
               0111151">Coherent states from combinatorial sequences</a>.
%H A006480 B. Salvy, <a href="http://algo.inria.fr/libraries/autocomb/AGM-html/AGM1.html">
               GFUN and the AGM</a>.
%H A006480 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BinomialSums.html">Link to a section of The World of Mathematics.</
               a>
%F A006480 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
%F A006480 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
%F A006480 a(n)=sum(k=-n, +n, (-1)^k*binomial(2*n, n+k)^3) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 02 2005
%F A006480 a(n)=C(2n,n)*C(3n,n)=A104684(2n,n); - Paul Barry (pbarry(AT)wit.ie), 
               Mar 14 2006
%p A006480 seq((3*n)!/(n!)^3, n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 28 2007
%t A006480 Sum [ (-1)^(k+n) Binomial[ 2n, k ]^3, {k, 0, 2n} ]
%o A006480 (PARI) a(n)=if(n<0,0,(3*n)!/n!^3)
%o A006480 (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))
%Y A006480 Cf. A000984, A050983, A050984, A008977.
%Y A006480 Sequence in context: A002432 A091800 A037959 this_sequence A138462 A002896 
               A004996
%Y A006480 Adjacent sequences: A006477 A006478 A006479 this_sequence A006481 A006482 
               A006483
%K A006480 nonn,easy,nice
%O A006480 0,2
%A A006480 N. J. A. Sloane (njas(AT)research.att.com).
%E A006480 More terms from Eric Weisstein (eric(AT)weisstein.com)

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