This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A177234 #9 Apr 28 2024 11:38:02
%S A177234 2,21,364,8855,278256,10737573,491796152,26088783435,1573664496040,
%T A177234 106395830418878,7970714909592876,655454164338881388,
%U A177234 58702034425556612832,5687847988198592380965,592867741295430227919600
%N A177234 a(n) = binomial(n^2, n)/(n+1).
%C A177234 Theorem: binomial(n^2, n)/(n+1) is an integer for n >= 2.
%C A177234 Proof 1 from William J. Keith, May 08 2010:
%C A177234 binomial(n^2, n) * 1/(n+1)
%C A177234 = (n^2)*(n^2-1)*(n^2-2)!/((n^2-n)!*n*(n-1)*(n-2)!) * 1/(n+1)
%C A177234 = n*(n^2-2)!/((n^2-n)!*(n-2)!) = n * binomial(n^2-2,n-2). QED
%C A177234 Proof 2 from Max Alekseyev, May 08 2010:
%C A177234 Recall that the valuation of m! w.r.t. prime p equals the sum floor(m/p^i) over i=1,2,3,...
%C A177234 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 A177234 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 A177234 It is clear that trouble may come only from primes dividing n+1.
%C A177234 Let valuation(n+1,p)=k > 0, i.e., n+1=p^k*m where prime p does not divide m.
%C A177234 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 A177234 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 A177234 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 A177234 G. C. Greubel, <a href="/A177234/b177234.txt">Table of n, a(n) for n = 2..335</a>
%H A177234 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]
%F A177234 From _G. C. Greubel_, Apr 27 2024: (Start)
%F A177234 a(n) = A014062(n)/(n+1).
%F A177234 a(n) = A177456(n)/(n-1).
%F A177234 a(n) = n*A177784(n).
%F A177234 a(n) = A177788(n)/n. (End)
%e A177234 a(3) = 21 because binomial(9,3)/(3+1) = 84/4 = 21.
%p A177234 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):
%t A177234 Table[Binomial[n^2,n]/(n+1), {n,2,30}] (* _G. C. Greubel_, Apr 27 2024 *)
%o A177234 (Magma)
%o A177234 [Binomial(n^2,n)/(n+1): n in [2..30]]; // _G. C. Greubel_, Apr 27 2024
%o A177234 (SageMath)
%o A177234 [binomial(n^2,n)/(n+1) for n in range(2,31)] # _G. C. Greubel_, Apr 27 2024
%Y A177234 Cf. A014062 A123312, A177456, A177784, A177788.
%K A177234 nonn
%O A177234 2,1
%A A177234 _Michel Lagneau_, May 05 2010, May 08 2010