%I #13 Sep 08 2022 08:45:15
%S 0,3,1,3,1,30,1,35,10,78,1,62,1,52,135,35,1,138,1,10,402,124,1,270,
%T 126,172,253,476,1,812,1,291,978,870,616,674,1,364,10,410,1,756,1,
%U 1124,1260,532,1,1422,1716,1128,2322,1556,1,1920,1941,2172,1815,844,1,3528,1,964
%N a(n) = binomial(2n-1,n-1) mod n^2.
%C For odd primes p, Charles Babbage showed in 1819 that a(p) = 1.
%H Chai Wah Wu, <a href="/A099906/b099906.txt">Table of n, a(n) for n = 1..10000</a>
%e a(11) = binomial(21,10) mod 11^2 = 352716 mod 121 = 1.
%t Table[ Mod[ Binomial[2n - 1, n - 1], n^2], {n, 60}] (* _Robert G. Wilson v_, Dec 14 2004 *)
%o (Magma) [Binomial(2*n-1, n-1) mod(n^2): n in [1..65]]; // _Vincenzo Librandi_, Jul 29 2015
%o (PARI) A099906(n)=binomial(2*n-1,n-1)%n^2 \\ _M. F. Hasler_, Jul 30 2015
%o (Python)
%o from __future__ import division
%o A099906_list, b = [], 1
%o for n in range(1,10001):
%o A099906_list.append(b % n**2)
%o b = b*2*(2*n+1)//(n+1) # _Chai Wah Wu_, Jan 26 2016
%Y Cf. A088218, A099905, A099907, A099908.
%K nonn
%O 1,2
%A _Henry Bottomley_, Oct 29 2004