

A008977


a(n) = (4*n)!/(n!)^4.


18



1, 24, 2520, 369600, 63063000, 11732745024, 2308743493056, 472518347558400, 99561092450391000, 21452752266265320000, 4705360871073570227520, 1047071828879079131681280, 235809301462142612780721600
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OFFSET

0,2


COMMENTS

Number of paths of length 4n 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 4way tie when 4n 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 3n 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


REFERENCES

N. Narumiya and H. Shiga, "The mirror map for a family of K3 surfaces induced from the simplest 3dimensional reflexive polytope", Proceedings on Moonshine and related topics (MontrĂ©al, QC, 1999), 139161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001. MR1877764 (2002m:14030)


LINKS

T. D. Noe, Table of n, a(n) for n=0..100
R. M. Dickau, Paths through a 4D lattice


FORMULA

a(n) = A139541(n)*(A001316(n)/A049606(n))^3.  Reinhard Zumkeller, Apr 28 2008
Selfconvolution of A178529, where A178529(n) = (4^n/n!^2) * Product_{k=0..n1} (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*n1/2) / (Pi*n)^(3/2).  Vaclav Kotesovec, Mar 07 2014
G.f.: hypergoem([1/4, 2/4, 3/4], [1, 1], 256*x).  Michael Somos, Aug 12 2014


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 4voter 3vote election that ends in a fourway 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


CROSSREFS

Cf. A000984, A006480, A008978, A178529.
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



