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A005790
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4-dimensional Catalan numbers.
(Formerly M4954)
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10
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1, 1, 14, 462, 24024, 1662804, 140229804, 13672405890, 1489877926680, 177295473274920, 22661585038594320, 3073259571003214320, 438091463242348309440, 65166105157299311029200, 10056663345892631910888600, 1602608179958939072505281850, 262708662267696303439658400600
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OFFSET
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0,3
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COMMENTS
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Number of standard tableaux of shape (n,n,n,n). - Emeric Deutsch, May 13 2004
The prime terms (as defined in A268538) are 1, 1, 10, 320, 16764, 1171355, 99315236, 9691755128, 1053114415100, ... - R. J. Mathar, Feb 27 2018
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Snover, Stephen L.; Troyer, Stephanie F.; A four-dimensional Catalan formula. Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 (1990), 123-126.
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LINKS
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FORMULA
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a(n) = 12*(4*n)!/(n! *(n+1)! *(n+2)! *(n+3)!).
G.f.: 4_F_3 ( [ 1, 3/2, 5/4, 7/4 ]; [ 3, 4, 5 ]; 256 x ).
(n+3)*(n+2)*(n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 04 2018
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MAPLE
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a:= n-> (4*n)! * mul(i!/(4+i)!, i=0..n-1):
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MATHEMATICA
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Table[12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!), {n, 0, 20}] (* Vaclav Kotesovec, Nov 18 2016 *)
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PROG
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(Magma) [12*Factorial(4*n)/(Factorial(n)*Factorial(n+1)*Factorial(n+2) *Factorial(n+3)): n in [0..20]]; // Vincenzo Librandi, Nov 23 2018
(PARI) vector(20, n, n--; 12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!)) \\ G. C. Greubel, Nov 23 2018
(Sage) [12*factorial(4*n)/(factorial(n)*factorial(n+1)*factorial(n+2) *factorial(n+3)) for n in range(20)] # G. C. Greubel, Nov 23 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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