This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A034841 a(n) = (n^2)! / (n!)^n. 16
 1, 1, 6, 1680, 63063000, 623360743125120, 2670177736637149247308800, 7363615666157189603982585462030336000, 18165723931630806756964027928179555634194028454000000, 53130688706387569792052442448845648519471103327391407016237760000000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The number of arrangements of 1,2,...,n*n in an n X n matrix such that each row is increasing. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 12 2001 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 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 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 LINKS Alois P. Heinz and Tilman Piesk, Table of n, a(n) for n = 0..26 (first 20 terms from Alois P. Heinz) FORMULA 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 a(n) = Product_{k=1..n} binomial(k*n, n). - Vaclav Kotesovec, Mar 10 2019 MAPLE a:= n-> (n^2)! / (n!)^n: seq(a(n), n=0..10);  # Alois P. Heinz, Jul 24 2012 MATHEMATICA 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 *) PROG (PARI) a(n) = (n^2)! / (n!)^n; \\ Michel Marcus, Oct 28 2014 (MAGMA) [Factorial(n^2) / Factorial(n)^n: n in [0..10]]; // Vincenzo Librandi, Oct 29 2014 CROSSREFS Cf. A000142, A039622, A229050, A229050. Diagonal of A089759, A187783. - Alois P. Heinz, Jan 23 2013 Sequence in context: A308029 A160226 A209609 * A149187 A330056 A258900 Adjacent sequences:  A034838 A034839 A034840 * A034842 A034843 A034844 KEYWORD nonn AUTHOR EXTENSIONS a(0)=1 prepended by Tilman Piesk, Oct 28 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 09:33 EST 2019. Contains 329843 sequences. (Running on oeis4.)