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A008977 a(n) = (4*n)!/(n!)^4.

%I A008977 #126 Feb 19 2025 13:01:30
%S A008977 1,24,2520,369600,63063000,11732745024,2308743493056,472518347558400,
%T A008977 99561092450391000,21452752266265320000,4705360871073570227520,
%U A008977 1047071828879079131681280,235809301462142612780721600,53644737765488792839237440000,12309355935372581458927646400000
%N A008977 a(n) = (4*n)!/(n!)^4.
%C A008977 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).
%C A008977 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
%C A008977 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
%C A008977 a(n) is the constant term of (X + Y + Z + 1/(X*Y*Z))^(4*n). - _Mark van Hoeij_, May 07 2013
%C A008977 In Narumiya and Shiga on page 158 the g.f. is given as a hypergeometric function. - _Michael Somos_, Aug 12 2014
%C A008977 Diagonal of the rational function R(x,y,z,w) = 1/(1-(w+x+y+z)). - _Gheorghe Coserea_, Jul 15 2016
%H A008977 T. D. Noe, <a href="/A008977/b008977.txt">Table of n, a(n) for n=0..100</a>
%H A008977 R. M. Dickau, <a href="https://www.robertdickau.com/path4d.html">Paths through a 4-D lattice</a>.
%H A008977 S. Hassani, J.-M. Maillard, and N. Zenine, <a href="https://arxiv.org/abs/2502.05543">On the diagonals of rational functions: the minimal number of variables (unabridged version)</a>, arXiv:2502.05543 [math-ph], 2025. See pp. 41.
%H A008977 Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
%H A008977 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 A008977 R. Mestrovic, <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 A008977 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 A008977 N. Narumiya and H. Shiga, <a href="https://www.researchgate.net/publication/268489728_The_mirror_map_for_a_family_of_K3_surfaces_induced_from_the_simplest_3-dimensional_reflexive_polytope">The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope</a>, Proceedings on Moonshine and related topics (Montréal, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001.  MR1877764 (2002m:14030).
%F A008977 a(n) = A139541(n)*(A001316(n)/A049606(n))^3. - _Reinhard Zumkeller_, Apr 28 2008
%F A008977 Self-convolution of A178529, where A178529(n) = (4^n/n!^2) * Product_{k=0..n-1} (8*k + 1)*(8*k + 3).
%F A008977 G.f.: hypergeom([1/8, 3/8], [1], 256*x)^2. - _Mark van Hoeij_, Nov 16 2011
%F A008977 a(n) ~ 2^(8*n - 1/2) / (Pi*n)^(3/2). - _Vaclav Kotesovec_, Mar 07 2014
%F A008977 G.f.: hypergeom([1/4, 2/4, 3/4], [1, 1], 256*x). - _Michael Somos_, Aug 12 2014
%F A008977 From _Peter Bala_, Jul 12 2016: (Start)
%F A008977 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)
%F A008977 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
%F A008977 From _Peter Bala_, Jul 17 2016: (Start)
%F A008977 a(n) = Sum_{k = 0..3*n} (-1)^(n+k)*binomial(4*n,n + k)* binomial(n + k,k)^4.
%F A008977 a(n) = Sum_{k = 0..4*n} (-1)^k*binomial(4*n,k)*binomial(n + k,k)^4. (End)
%F A008977 E.g.f.: 3F3(1/4,1/2,3/4; 1,1,1; 256*x). - _Ilya Gutkovskiy_, Jan 23 2018
%F A008977 From _Peter Bala_, Feb 16 2020: (Start)
%F A008977 a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
%F A008977 a(n) = [(x*y*z)^n] (1 + x + y + z)^(4*n). (End)
%F A008977 D-finite with recurrence n^3*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - _R. J. Mathar_, Aug 01 2022
%F A008977 a(n) = 24*A082368(n). - _R. J. Mathar_, Jun 21 2023
%e A008977 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
%e A008977 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
%e A008977 G.f. = 1 + 24*x + 2520*x^2 + 369600*x^3 + 63063000*x^4 + 11732745024*x^5 + ...
%p A008977 A008977 := n->(4*n)!/(n!)^4;
%t A008977 Table[(4n)!/(n!)^4,{n,0,16}] (* _Harvey P. Dale_, Oct 24 2011 *)
%t A008977 a[ n_] := If[ n < 0, 0, (4 n)! / n!^4]; (* _Michael Somos_, Aug 12 2014 *)
%t A008977 a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/4, 2/4, 3/4}, {1, 1}, 256 x], {x, 0, n}]; (* _Michael Somos_, Aug 12 2014 *)
%o A008977 (Maxima) A008977(n):=(4*n)!/(n!)^4$ makelist(A008977(n),n,0,20); /* _Martin Ettl_, Nov 15 2012 */
%o A008977 (Magma) [Factorial(4*n)/Factorial(n)^4: n in [0..20]]; // _Vincenzo Librandi_, Aug 13 2014
%o A008977 (PARI) a(n) = (4*n)!/n!^4; \\ _Gheorghe Coserea_, Jul 15 2016
%o A008977 (Python)
%o A008977 from math import factorial
%o A008977 def A008977(n): return factorial(n<<2)//factorial(n)**4 # _Chai Wah Wu_, Mar 15 2023
%Y A008977 Cf. A000984, A006480, A008978, A178529, A000897, A002894, A002897, A006480, A008979, A186420, A188662.
%Y A008977 Row 4 of A187783.
%Y A008977 Related to diagonal of rational functions: A268545-A268555.
%K A008977 nonn,easy,nice
%O A008977 0,2
%A A008977 _N. J. A. Sloane_