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A006480 De Bruijn's S(3,n): (3n)!/(n!)^3.
(Formerly M4284)
54
1, 6, 90, 1680, 34650, 756756, 17153136, 399072960, 9465511770, 227873431500, 5550996791340, 136526995463040, 3384731762521200, 84478098072866400, 2120572665910728000, 53494979785374631680, 1355345464406015082330 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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

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

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

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

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

Diagonal of the rational function 1/(1 - x - y - z). - Gheorghe Coserea, Jul 06 2016

REFERENCES

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

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.

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

LINKS

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

G. E. Andrews, The well-poised thread: An Organized Chronicle of Some Amazing Summations and their Implications, Ramanujan J., 1 (1997), 7-23; see Section 8.

A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

H. J. Brothers, Pascal's Prism: Supplementary Material.

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

H. W. Gould, Tables of Combinatorial Identities, Edited by J. Quaintance.

B. Hinman, Letter to N. J. A. Sloane, Aug. 1980.

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.

Pedro J. Miana, Hideyuki Ohtsuka, Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

T. Peck, Letter to N. J. A. Sloane, Aug. 1980.

K. A. Penson and A. I. Solomon, Coherent states from combinatorial sequences, arXiv:quant-ph/0111151, 2001.

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

B. Salvy, GFUN and the AGM.

R. Sprugnoli, Riordan array proofs of identities in Gould's book.

Dennis Walsh, The probability of a run-off election when three equally-favored candidates vie for two slots.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups".

Eric Weisstein's World of Mathematics, Binomial Sums.

Wikipedia, Dixon's identity.

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 A. Penson, Nov 21 2001

a(n) = Sum_{k=-n..n} (-1)^k*binomial(2*n, n+k)^3. - Benoit Cloitre, Mar 02 2005

a(n) = C(2n,n)*C(3n,n) = A104684(2n,n). - Paul Barry, Mar 14 2006

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

n^2*a(n) - 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Dec 04 2012

a(n) = (n+1)^2*(n+2)*A161581(n) for n>2. - Alexander Adamchuk, Dec 27 2013

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

0 = x*(27*x-1)*y'' + (54*x-1)*y' + 6*y, where y is g.f. - Gheorghe Coserea, Jul 06 2016

From Peter Bala, Jul 15 2016: (Start)

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.

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.

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

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

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

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

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

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

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).

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)

EXAMPLE

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

MAPLE

seq((3*n)!/(n!)^3, n=0..16); # Zerinvary Lajos, Jun 28 2007

MATHEMATICA

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

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

Table[Multinomial[n, n, n], {n, 0, 100}] (* Emanuele Munarini, Oct 25 2016 *)

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

(Maxima) makelist(multinomial_coeff(n, n, n), n, 0, 24); /* Emanuele Munarini, Oct 25 2016 */

CROSSREFS

Cf. A000984, A008977, A050983, A050984, A161581, A181545.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A037959 A247150 A201073 * A138462 A002896 A266734

Adjacent sequences:  A006477 A006478 A006479 * A006481 A006482 A006483

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Eric W. Weisstein

STATUS

approved

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Last modified October 20 14:04 EDT 2017. Contains 293601 sequences.