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A008977 a(n) = (4*n)!/(n!)^4. 25
1, 24, 2520, 369600, 63063000, 11732745024, 2308743493056, 472518347558400, 99561092450391000, 21452752266265320000, 4705360871073570227520, 1047071828879079131681280, 235809301462142612780721600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of paths of length 4*n in an n X n X n X n grid from (0,0,0,0) to (n,n,n,n).

a(n) occurs in Ramanujan's formula 1/Pi = sqrt(8)/9801 * Sum_{n>=0} (4*n)!/(n!)^4 * (1103 + 26390*n)/396^(4*n) ). - Susanne Wienand, Jan 05 2013

a(n) is the number of ballot results that lead to a 4-way tie when 4*n voters each cast three votes for three out of four candidates vying for 3 slots on a county commission; each of these ballot results give 3*n votes to each of the four candidates. - Dennis P. Walsh, May 02 2013

a(n) is the constant term of (X + Y + Z + 1/(X*Y*Z))^(4*n). - Mark van Hoeij, May 07 2013

In Narumiya and Shiga on page 158 the g.f. is given as a hypergeometric function. - Michael Somos, Aug 12 2014

Diagonal of the rational function R(x,y,z,w) = 1/(1-(w+x+y+z)). - Gheorghe Coserea, Jul 15 2016

LINKS

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

R. M. Dickau, Paths through a 4-D lattice

N. Narumiya and H. Shiga, The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope, Proceedings on Moonshine and related topics (MontrĂ©al, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001.  MR1877764 (2002m:14030).

FORMULA

a(n) = A139541(n)*(A001316(n)/A049606(n))^3. - Reinhard Zumkeller, Apr 28 2008

Self-convolution of A178529, where A178529(n) = (4^n/n!^2) * Product_{k=0..n-1} (8*k + 1)*(8*k + 3).

G.f.: hypergeom([1/8, 3/8], [1], 256*x)^2. - Mark van Hoeij, Nov 16 2011

a(n) ~ 2^(8*n - 1/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Mar 07 2014

G.f.: hypergeom([1/4, 2/4, 3/4], [1, 1], 256*x). - Michael Somos, Aug 12 2014

From Peter Bala, Jul 12 2016: (Start)

a(n) = binomial(2*n,n)*binomial(3*n,n)*binomial(4*n,n) = ( [x^n](1 + x)^(2*n) ) * ( [x^n](1 + x)^(3*n) ) * ( [x^n](1 + x)^(4*n) ) = [x^n](F(x)^(24*n)), where F(x) = 1 + x + 29*x^2 + 2246*x^3 + 239500*x^4 + 30318701*x^5 + 4271201506*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008978, A008979, A186420 and A188662. (End)

0 = (x^2-256*x^3)*y''' + (3*x-1152*x^2)*y'' + (1-816*x)*y' - 24*y, where y is the g.f. - Gheorghe Coserea, Jul 15 2016

From Peter Bala, Jul 17 2016: (Start)

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

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

EXAMPLE

a(13)=52!/(13!)^4=53644737765488792839237440000 is the number of ways of dealing the four hands in Bridge or Whist. - Henry Bottomley, Oct 06 2000

a(1)=24 since, in a 4-voter 3-vote election that ends in a four-way tie for candidates A, B, C, and D, there are 4! ways to arrange the needed vote sets {A,B,C}, {A,B,D}, {A,C,D}, and {B,C,D} among the 4 voters. - Dennis P. Walsh, May 02 2013

G.f. = 1 + 24*x + 2520*x^2 + 369600*x^3 + 63063000*x^4 + 11732745024*x^5 + ...

MAPLE

A008977 := n->(4*n)!/(n!)^4;

MATHEMATICA

Table[(4n)!/(n!)^4, {n, 0, 16}] (* Harvey P. Dale, Oct 24 2011 *)

a[ n_] := If[ n < 0, 0, (4 n)! / n!^4]; (* Michael Somos, Aug 12 2014 *)

a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/4, 2/4, 3/4}, {1, 1}, 256 x], {x, 0, n}]; (* Michael Somos, Aug 12 2014 *)

PROG

(Maxima) A008977(n):=(4*n)!/(n!)^4$ makelist(A008977(n), n, 0, 20); /* Martin Ettl, Nov 15 2012 */

(MAGMA) [Factorial(4*n)/Factorial(n)^4: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

(PARI) a(n) = (4*n)!/n!^4; \\ Gheorghe Coserea, Jul 15 2016

CROSSREFS

Cf. A000984, A006480, A008978, A178529, A000897, A002894, A002897, A006480, A008979, A186420, A188662.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A107675 A173115 A202927 * A159392 A064596 A217971

Adjacent sequences:  A008974 A008975 A008976 * A008978 A008979 A008980

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 10 09:47 EST 2016. Contains 278999 sequences.