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%I #4 Mar 30 2012 18:35:53
%S 2,21,364,8855,278256,10737573,491796152,26088783435,1573664496040,
%T 106395830418878,7970714909592876,655454164338881388,
%U 58702034425556612832,5687847988198592380965,592867741295430227919600
%N a(n) = binomial(n^2, n)/(n+1).
%C Theorem: binomial(n^2, n)/(n+1) is an integer for n >= 2.
%C Proof 1 from William J. Keith, May 08 2010:
%C binomial(n^2, n) * 1/(n+1)
%C = (n^2)(n^2-1)(n^2-2)!/((n^2-n)!n(n-1)(n-2)!) * 1/(n+1)
%C = n (n^2-2)!/((n^2-n)!(n-2)!) = n * binomial(n^2-2,n-2). QED
%C Proof 2 from Max Alekseyev, May 08 2010:
%C Recall that the valuation of m! w.r.t. prime p equals the sum floor(m/p^i) over i=1,2,3,...
%C Moreover, if m=a+b where a and b are nonnegative integers, then floor(m/p^i) - floor(a/p^i) - floor(b/p^i) >= 0.
%C Let n>1. To prove that binomial(n^2, n)/(n+1) is an integer, it is enough to show that its valuation w.r.t. any prime p is nonnegative.
%C It is clear that trouble may come only from primes dividing n+1.
%C Let valuation(n+1,p)=k > 0, i.e., n+1=p^k*m where prime p does not divide m.
%C Then n = p^k*m - 1, n^2 = p^(2k)*m^2 - 2*p^k*m + 1 and n^2 - n = p^(2k)*m^2 - 3*p^k*m + 2.
%C It is easy to check that floor(n^2/p^i) - floor(n/p^i) - floor((n^2-n)/p^i) = 1 for i=1,2,...,k if p>2 and for i=2,3,...,k+1 if p=2, implying that valuation(binomial(n^2, n)/(n+1),p) >= 0. QED
%D H. Gupta and S. P. Khare, On C(k^2,k) and the product of the first k primes, Publ. Fac. Electrotechn. Belgrade, Ser. Math. Phys. 25-29 (1977) 577-598.
%H H. Gupta and S. P. Khare, <a href="http://www.ams.org/mathscinet-getitem?mr=453615">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. [<a href="http://pefmath2.etf.rs/files/101/580.pdf">PDF</a>] [From _R. J. Mathar_, May 09 2010]
%e a(3) = 21 because binomial(9,3)/(3+1) = 84/4 = 21.
%p with(numtheory):n0:=25:T:=array(1..n0-1):for n from 2 to n0 do: T[n-1]:= binomial(n*n,n)/(n+1):od:print(T):
%Y Cf. A014062 A123312
%K nonn
%O 2,1
%A _Michel Lagneau_, May 05 2010, May 08 2010
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