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A039622
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Number of n X n Young tableaux.
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7
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1, 1, 2, 42, 24024, 701149020, 1671643033734960, 475073684264389879228560, 22081374992701950398847674830857600, 220381378415074546123953914908618547085974856000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of arrangements of 1,2,..,n^2 in an n X n matrix such that each row and each column is increasing.
Factor g_n in formula for 2n-th moment of Riemann zeta function on the critical line. (See Conrey article.) - Michael Somos, Apr 15 2003
Number of linear extensions of the n X n lattice. - Mitch Harris (Harris.Mitchell (AT) mgh.harvard.edu), Dec 27, 2005
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REFERENCES
| The problem for a 5 X 5 array was recently posed and solved in the College Mathematics Journal. The solution is in Vol. 30 (1999), no. 5, pp. 410-411.
J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 349.
J. S. Frame, G. de B. Robinson and R. M. Thrall, The hook graphs of a symmetric group, Canad. J. Math. 6 (1954), pp. 316-324.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 284.
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LINKS
| J. B. Conrey, The Riemann Hypothesis
P.-O. Dehaye, Combinatorics of the lower order terms in the moment conjectures: the Riemann zeta function
Index entries for sequences related to Young tableaux.
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FORMULA
| a(n) = (n^2)! / (product k=1, ..., 2n-1 k^(n - |n-k|))
a(n) = 0!*1!*..*(k-1)! *(k*n)! / ( n!*(n+1)!*..*(n+k-1)! ) for k=n.
a(n) = A088020(n)/A107254(n) = A088020(n)*A000984(n)/A079478(n). - Henry Bottomley (se16(AT)btinternet.com), May 14 2005
a(n) = A153452(prime(n)^n).- Naohiro Nomoto, Jan 01 2009
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EXAMPLE
| Using the hook length formula, a(4) = (16)!/(7*6^2*5^3*4^4*3^3*2^2) = 24024.
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MATHEMATICA
| a[n_] := (n^2)!*Product[ k!/(n + k)!, {k, 0, n - 1}]; Table[ a[n], {n, 0, 9}] (* From Jean-François Alcover, Dec 06 2011, after Pari *)
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PROG
| (PARI) a(n)=if(n<0, 0, (n^2)!*prod(k=0, n-1, k!/(n+k)!))
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CROSSREFS
| Main diagonal of A060854. a(2)=A000108(2), a(3)=A005789(3), a(4)=A005790(4), a(5)=A005791(5).
Sequence in context: A193271 A193272 A193273 * A130506 A052078 A069544
Adjacent sequences: A039619 A039620 A039621 * A039623 A039624 A039625
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Floor van Lamoen (fvlamoen(AT)hotmail.com)
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EXTENSIONS
| References, correction and extension from Stephen G. Penrice (spenrice(AT)ets.org), Jun 15 2000
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