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%I #67 Jul 14 2023 10:15:30
%S 1,1,6,1680,63063000,623360743125120,2670177736637149247308800,
%T 7363615666157189603982585462030336000,
%U 18165723931630806756964027928179555634194028454000000,53130688706387569792052442448845648519471103327391407016237760000000000
%N a(n) = (n^2)! / (n!)^n.
%C The number of arrangements of 1,2,...,n^2 in an n X n matrix such that each row is increasing. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001
%C a(n) == 0 mod (n!). In fact (n^2)! == 0 mod (n!)^n by elementary combinatorics, a better result is (n^2)! == 0 ((mod(n!)^(n+1)). - _Amarnath Murthy_, Jul 13 2005
%C a(n) is also the number of lattice paths from {n}^n to {0}^n using steps that decrement one component by 1. a(2) = 6: [(2,2), (1,2), (0,2), (0,1), (0,0)], [(2,2), (1,2), (1,1), (0,1), (0,0)], [(2,2), (1,2), (1,1), (1,0), (0,0)], [(2,2), (2,1), (1,1), (0,1), (0,0)], [(2,2), (2,1), (1,1), (1,0), (0,0)], [(2,2), (2,1), (2,0), (1,0), (0,0)]. - _Alois P. Heinz_, May 06 2013
%C Given n^2 distinguishable balls and n distinguishable urns, a(n) = the number of ways to place n balls in the i-th urn for all 1 <= i <= n, where n = n_1 + n_2 + ... + n_n. - _Ross La Haye_, Dec 28 2013
%H Alois P. Heinz and Tilman Piesk, <a href="/A034841/b034841.txt">Table of n, a(n) for n = 0..26</a> (first 20 terms from Alois P. Heinz)
%F Using a higher order version of Stirling's formula (the "standard" formula appears in A000142) we have the asymptotic expression: a(n) ~ sqrt(2*Pi) * e^(-1/12) * n^(n^2 - n/2 + 1) / (2*Pi)^(n/2). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001
%F a(n) = Product_{k=1..n} binomial(k*n, n). - _Vaclav Kotesovec_, Mar 10 2019
%p a:= n-> (n^2)! / (n!)^n:
%p seq(a(n), n=0..10); # _Alois P. Heinz_, Jul 24 2012
%t Prepend[Table[nn = n^2;nn! Coefficient[Series[(x^n/n!)^n, {x, 0, nn}], x^nn], {n, 1, 15}], 1] (* _Geoffrey Critzer_, Mar 08 2015 *)
%o (PARI) a(n) = (n^2)! / (n!)^n; \\ _Michel Marcus_, Oct 28 2014
%o (Magma) [Factorial(n^2) / Factorial(n)^n: n in [0..10]]; // _Vincenzo Librandi_, Oct 29 2014
%Y Cf. A000142, A039622, A229050, A229050, A340590.
%Y Cf. A000984, A006480, A008977, A008978, A022915.
%Y Main diagonal of A089759, A187783.
%K nonn
%O 0,3
%A _Erich Friedman_
%E a(0)=1 prepended by _Tilman Piesk_, Oct 28 2014
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