%I M4284 #245 Nov 07 2024 03:52:12

%S 1,6,90,1680,34650,756756,17153136,399072960,9465511770,227873431500,

%T 5550996791340,136526995463040,3384731762521200,84478098072866400,

%U 2120572665910728000,53494979785374631680,1355345464406015082330,34469858696831179429500,879619727485803060256500,22514366432046593564460000

%N De Bruijn's S(3,n): (3n)!/(n!)^3.

%C Number of paths of length 3n in an n X n X n grid from (0,0,0) to (n,n,n), using steps (0,0,1), (0,1,0), and (1,0,0).

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

%C 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

%C a(n) is the number of ballot results that end in a 3-way tie when 3n voters each cast two votes for two out of three candidates vying for 2 slots on a county board; in such a tie, each of the three candidates receives 2n votes. Note there are C(3n,2n) ways to choose the voters who cast a vote for the youngest candidate. The n voters who did note vote for the youngest candidate voted for the two older candidates. Then there are C(2n,n) ways to choose the other n voters who voted for both the youngest and the second youngest candidate. The remaining voters vote for the oldest candidate. Hence there are C(3n,2n)*C(2n,n)=(3n)!/(n!)^3 ballot results. - _Dennis P. Walsh_, May 02 2013

%C a(n) is the constant term of (X+Y+1/(X*Y))^(3*n). - _Mark van Hoeij_, May 07 2013

%C For n > 2 a(n) is divisible by (n+2)*(n+1)^2, a(n) = (n+1)^2*(n+2)*A161581(n). - _Alexander Adamchuk_, Dec 27 2013

%C a(n) is the number of permutations of the multiset {1^n, 2^n, 3^n}, the number of ternary words of length 3*n with n of each letters. - _Joerg Arndt_, Feb 28 2016

%C Diagonal of the rational function 1/(1 - x - y - z). - _Gheorghe Coserea_, Jul 06 2016

%C At least two families of elliptic curves, x = 2*H1 = (p^2+q^2)*(1-q) and x = 2*H2 = p^2+q^2-3*p^2*q+q^3 (0<x<4/27), generate this sequence via the period-energy function T(x) = 2*Pi*2F1(1/3,2/3; 1; (27/4)*x). - _Bradley Klee_, Feb 25 2018

%C The ordinary generating function also determines periods along a family of tetrahedral-symmetric sphere curves ("du troisième ordre"). Compare links to Goursat "Étude des surfaces..." and "Proof Certificate". - _Bradley Klee_, Sep 28 2018

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

%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 174.

%D N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publishing Co., 1958. See chapters 4 and 6.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006480/b006480.txt">Table of n, a(n) for n = 0..100</a>

%H George E. Andrews, <a href="http://dx.doi.org/10.1023/A:1009746303654">The well-poised thread: An Organized Chronicle of Some Amazing Summations and their Implications</a>, Ramanujan J., 1 (1997), 7-23; see Section 8.

%H A. Bostan, S. Boukraa, J.-M. Maillard and J.-A. Weil, <a href="https://doi.org/10.1088/1751-8113/48/50/504001">Diagonals of rational functions and selected differential Galois groups</a>, Journal of Physics A: Mathematical and Theoretical, Vol. 48, No. 50 (2015), 504001; <a href="http://arxiv.org/abs/1507.03227">arXiv preprint</a>, arXiv:1507.03227 [math-ph], 2015.

%H Alin Bostan, Armin Straub, and Sergey Yurkevich, <a href="https://arxiv.org/abs/2212.10116">On the representability of sequences as constant terms</a>, arXiv:2212.10116 [math.NT], 2022.

%H Harlan J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>.

%H Robert M. Dickau, <a href="https://www.robertdickau.com/path3d.html">3-dimensional shortest-path diagrams</a>.

%H Henry W. Gould, <a href="https://web.archive.org/web/20190411015737/https://www.math.wvu.edu/~gould/">Tables of Combinatorial Identities</a>, Edited by J. Quaintance.

%H Édouard Goursat, <a href="http://www.numdam.org/item/ASENS_1887_3_4__159_0">Étude des surfaces qui admettent tous les plans de symétrie d'un polyèdre régulier</a>, Annales scientifiques de l'École Normale Supérieure, Série 3 : Volume 4 (1887), 165-166.

%H Bob Hinman, <a href="/A006480/a006480_1.pdf">Letter to N. J. A. Sloane, Aug. 1980</a>.

%H Brad Klee, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-December/018186.html">Geometric G.F. for Ramanujan Periods</a>, seqfans mailing list, 2017.

%H Bradley Klee, <a href="/A006480/a006480_2.pdf">Proof Certificate</a>.

%H Markus Kuba and Alois Panholzer, <a href="https://arxiv.org/abs/2411.03930">Lattice paths and the diagonal of the cube</a>, arXiv:2411.03930 [math.CO], 2024.

%H Gilbert Labelle and Annie Lacasse, <a href="https://hal.inria.fr/hal-01215040/">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavík, Iceland DMTCS proc. AO, 2011, pp. 599-610.

%H Romeo Meštrović, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.

%H Pedro J. Miana, Hideyuki Ohtsuka, and Natalia Romero, <a href="https://doi.org/10.1016/j.disc.2017.05.006">Sums of powers of Catalan triangle numbers</a>, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2388-2397; <a href="http://arxiv.org/abs/1602.04347">arXiv preprint</a>, arXiv:1602.04347 [math.NT], 2016.

%H Jovan Mikić, <a href="https://www.emis.de/journals/JIS/VOL21/Mikic/mikic25.html">A Method For Examining Divisibility Properties Of Some Binomial Sums</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.7.

%H Michaël Moortgat, <a href="https://cla.tcs.uj.edu.pl/history/2020/pdfs/CLA_Moortgat.pdf">The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus</a>, 15th Workshop: Computational Logic and Applications (CLA 2020).

%H Trey Peck, <a href="/A006480/a006480.pdf">Letter to N. J. A. Sloane, Aug. 1980</a>.

%H Karol A. Penson and Allan I. Solomon, <a href="https://doi.org/10.1142/9789812777850_0066">Coherent states from combinatorial sequences</a>, in: E. Kapuscik and A. Horzela (eds.), Quantum theory and symmetries, World Scientific, 2002, pp. 527-530; <a href="https://arxiv.org/abs/quant-ph/0111151">arXiv preprint</a>, arXiv:quant-ph/0111151, 2001.

%H Marko Petkovsek, Herbert Wilf and Doron Zeilberger, <a href="https://www.math.upenn.edu/~wilf/AeqB.html">A=B</a>, A K Peters, 1996, p. 22.

%H Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf">Modular Equations and Approximations to Pi</a>, Quarterly Journal of Mathematics, XLV (1914), 350-372.

%H Bruno Salvy, <a href="http://algo.inria.fr/libraries/autocomb/AGM-html/AGM1.html">GFUN and the AGM</a>.

%H Renzo Sprugnoli, <a href="https://web.archive.org/web/20120127210623/http://www.dsi.unifi.it/~resp/GouldBK.pdf">Riordan array proofs of identities in Gould's book</a>, 2006.

%H Dennis P. Walsh, <a href="https://web.archive.org/web/20221006050044/http://capone.mtsu.edu/dwalsh/VOTENEW.pdf">The probability of a run-off election when three equally-favored candidates vie for two slots</a>.

%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dixon%27s_identity">Dixon's identity</a>.

%F 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 From _Karol A. Penson_, Nov 21 2001: (Start)

%F O.g.f.: hypergeom([1/3, 2/3], [1], 27*x).

%F E.g.f.: hypergeom([1/3, 2/3], [1, 1], 27*x).

%F Integral representation as n-th moment of a positive function on [0, 27]:

%F 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). This representation is unique. (End)

%F a(n) = Sum_{k=-n..n} (-1)^k*binomial(2*n, n+k)^3. - _Benoit Cloitre_, Mar 02 2005

%F a(n) = C(2n,n)*C(3n,n) = A104684(2n,n). - _Paul Barry_, Mar 14 2006

%F G.f. satisfies: A(x^3) = A( x*(1+3*x+9*x^2)/(1+6*x)^3 )/(1+6*x). - _Paul D. Hanna_, Oct 29 2010

%F D-finite with recurrence: n^2*a(n) - 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - _R. J. Mathar_, Dec 04 2012

%F a(n) = (n+1)^2*(n+2)*A161581(n) for n>2. - _Alexander Adamchuk_, Dec 27 2013

%F 0 = a(n)^2*(472392*a(n+1)^2 - 83106*a(n+1)*a(n+2) + 3600*a(n+2)^2) + a(n)*a(n+1)*(-8748*a(n+1)^2 + 1953*a(n+1)*a(n+2) - 120*a(n+2)^2) + a(n+1)^2*(+36*a(n+1)^2 - 12*a(n+1)*a(n+2) + a(n+2)^2 for all n in Z. - _Michael Somos_, Oct 22 2014

%F 0 = x*(27*x-1)*y'' + (54*x-1)*y' + 6*y, where y is g.f. - _Gheorghe Coserea_, Jul 06 2016

%F From _Peter Bala_, Jul 15 2016: (Start)

%F a(n) = 3*binomial(2*n - 1,n)*binomial(3*n - 1,n) = 3*[x^n] 1/(1 - x)^n * [x^n] 1/(1 - x)^(2*n) for n >= 1.

%F a(n) = binomial(2*n,n)*binomial(3*n,n) = ([x^n](1 + x)^(2*n)) *([x^n](1 + x)^(3*n)) = [x^n](F(x)^(6*n)), where F(x) = 1 + x + 2*x^2 + 14*x^3 + 127*x^4 + 1364*x^5 + 16219*x^6 + ... appears to have integer coefficients. Cf. A002894.

%F This sequence occurs as the right-hand side of several binomial sums:

%F Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n,k)^3 = a(n) (Dixon's identity).

%F Sum_{k = 0..n} binomial(n,k)*binomial(2*n,n - k)*binomial(3*n + k,k) = a(n) (Gould, Vol. 4, 6.86)

%F Sum_{k = 0..n} (-1)^(n+k)*binomial(n,k)*binomial(2*n + k,n)*binomial(3*n + k,n) = a(n).

%F Sum_{k = 0..n} binomial(n,k)*binomial(2*n + k,k)*binomial(3*n,n - k) = a(n).

%F Sum_{k = 0..n} (-1)^(k)*binomial(n,k)*binomial(3*n - k,n)*binomial(4*n - k,n) = a(n).

%F Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n + k,2*n - k)*binomial(2*k,k)*binomial(4*n - k,2*n) = a(n) (see Gould, Vol.5, 9.23).

%F Sum_{k = 0..2*n} (-1)^k*binomial(3*n,k)*binomial(3*n - k,n)^3 = a(n) (Sprugnoli, Section 2.9, Table 10, p. 123). (End)

%F From _Bradley Klee_, Feb 28 2018: (Start)

%F a(n) = A005809(n)*A000984(n).

%F G.f.: F(x) = 1/(2*Pi) Integral_{z=0..2*Pi} 2F1(1/3,2/3; 1/2; 27*x*sin^2(z)) dz.

%F With G(x) = x*2F1(1/3,2/3; 2; 27*x): F(x) = d/dx G(x). (Cf. A007004) (End)

%F F(x)*G(1/27-x) + F(1/27-x)*G(x) = 1/(4*Pi*sqrt(3)). - _Bradley Klee_, Sep 29 2018

%F Sum_{n>=0} 1/a(n) = A091683. - _Amiram Eldar_, Nov 15 2020

%F From _Peter Bala_, Sep 20 2021: (Start)

%F a(n) = Sum_{k = n..2*n} binomial(2*n,k)^2 * binomial(k,n). Cf. A001459.

%F a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive integers n and k (write a(n) as C(3*n,2*n)*C(2*n,n) and apply Mestrovic, equation 39, p. 12). (End)

%F a(n) = 6*A060542(n). - _R. J. Mathar_, Jun 21 2023

%F Occurs on the right-hand side of the binomial sum identities Sum_{k = -n..n} (-1)^k * (n + x - k) * binomial(2*n, n+k)^3 = (x + n)*a(n) and Sum_{k = -n..n} (-1)^k * (n + x - k)^3 * binomial(2*n, n+k)^3 = x*(x + n)*(x + 2*n)*a(n) (x arbitrary). Compare with Dixon's identity: Sum_{k = -n..n} (-1)^k * binomial(2*n, n+k)^3 = a(n). - _Peter Bala_, Jul 31 2023

%F From _Peter Bala_, Aug 14 2023: (Start)

%F a(n) = (-1)^n * [x^(2*n)] ( (1 - x)^(4*n) * Legendre_P(2*n, (1 + x)/(1 - x)) ).

%F Row 1 of A364509. (End)

%F From _Peter Bala_, Oct 10 2024: (Start)

%F The following hold for n >= 1:

%F a(n) = Sum_{k = 0.. 2*n} (-1)^(n+k) * k/n * binomial(2*n, k)^3 = 3/2 * Sum_{k = 0.. 2*n} (-1)^(n+k) * (k/n)^2 * binomial(2*n, k)^3.

%F a(n) = 3/2 * Sum_{0..2*n-1} (-1)^(n+k) * k/n * binomial(2*n, k)^2*binomial(2*n-1, k).

%F a(n) = 3 * Sum_{0..2*n-1} (-1)^(n+k) * k/n * binomial(2*n, k)*binomial(2*n-1, k)^2. (End)

%F a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k) * A108625(2*n, k) (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). - _Peter Bala_, Oct 15 2024

%e G.f.: 1 + 6*x + 90*x^2 + 1680*x^3 + 34650*x^4 + 756756*x^5 + 17153136*x^6 + ...

%p seq((3*n)!/(n!)^3, n=0..16); # _Zerinvary Lajos_, Jun 28 2007

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

%t a[ n_] := If[ n < 0, 0, (-1)^n HypergeometricPFQ[ {-2 n, -2 n, -2 n}, {1, 1}, 1]]; (* _Michael Somos_, Oct 22 2014 *)

%t Table[Multinomial[n, n, n], {n, 0, 100}] (* _Emanuele Munarini_, Oct 25 2016 *)

%t CoefficientList[Series[Hypergeometric2F1[1/3,2/3,1,27*x],{x,0,5}],x] (* _Bradley Klee_, Feb 28 2018 *)

%o (PARI) {a(n) = if( n<0, 0, (3*n)! / n!^3)}; /* _Michael Somos_, Dec 03 2002 */

%o (PARI) {a(n) = my(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))}; /* _Michael Somos_, Dec 03 2002 */

%o (Magma) [Factorial(3*n)/(Factorial(n))^3: n in [0..20] ]; // _Vincenzo Librandi_, Aug 20 2011

%o (Maxima) makelist(multinomial_coeff(n,n,n),n,0,24); /* _Emanuele Munarini_, Oct 25 2016 */

%o (GAP) List([0..20],n->Factorial(3*n)/Factorial(n)^3); # _Muniru A Asiru_, Mar 31 2018

%o (Python)

%o from math import factorial

%o def A006480(n): return factorial(3*n)//factorial(n)**3 # _Chai Wah Wu_, Oct 04 2022

%Y Cf. A000984, A001459, A008977, A050983, A050984, A091683, A161581, A181545.

%Y Row 3 of A187783.

%Y Related to diagonal of rational functions: A268545-A268555. Elliptic Integrals: A002894, A113424, A000897. Factors: A005809, A000984. Integrals: A007004, A024486. Sphere Curves: A318245, A318495.

%K nonn,easy,nice,changed

%O 0,2

%A _N. J. A. Sloane_

%E a(14)-a(16) from _Eric W. Weisstein_

%E Terms a(17) and beyond from _T. D. Noe_, Jun 29 2008