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%I #51 Mar 25 2024 12:05:18
%S 1,1,6,84,1820,53130,1947792,85900584,4426165368,260887834350,
%T 17310309456440,1276749965026536,103619293824707388,
%U 9176358300744339432,880530516383349192480,91005567811177478095440,10078751602022313874633200,1190739044344491048895397910
%N a(n) = binomial(n^2, n).
%C Roberts states that Gupta and Khare show that a(n) > A002110(n) for 2 < n < 1794 and that a(n) < A002110(n) for n >= 1794, where A002110(n) is the product of the first n primes. - _T. D. Noe_, Oct 03 2007
%C This sequence describes the number of ways to arrange n objects in an n X n array (for example, stars in a flag's field pattern). - Tom Young (mcgreg265(AT)msn.com), Jun 17 2010
%C It appears that a(n) == n (mod n^3) only if n is 1, an odd prime, the square of an odd prime, or the cube of an odd prime. - _Gary Detlefs_, Aug 06 2013; corrected by _Michel Marcus_, May 29 2015
%D J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 265.
%H T. D. Noe, <a href="/A014062/b014062.txt">Table of n, a(n) for n=0..100</a>
%H H. Alzer and J. Sandor, <a href="https://dx.doi.org/10.2298/AADM110206008A">On a binomial coefficient and a product of prime numbers</a>, Appl. An. Disc. Math. 5 (2011) 87-92.
%H H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>.
%H H. Gupta and S. P. Khare, <a href="http://pefmath2.etf.rs/files/101/580.pdf">On C(k^2,k) and the product of the first k primes</a>, Publ. Fac. Electrotechn. Belgrade, Ser. Math. Phys. 25-29 (1977) 577-598.
%F a(n) ~ 1/sqrt(2*Pi) * (e*n)^(n - 1/2). - _Charles R Greathouse IV_, Jul 07 2007
%F a(n) = Sum_{k=0..n} binomial(n, k) * binomial(n^2 - n, k). - _Paul D. Hanna_, Nov 18 2015
%F a(n) = (n+1)*A177234(n). - _R. J. Mathar_, Jan 25 2019
%t Table[Binomial[n^2,n],{n,0,22}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 03 2011 *)
%o (PARI) {a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2-n, k))}
%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 18 2015
%Y Main diagonal of A060539.
%Y Cf. A295773.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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