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%I #18 Mar 19 2024 09:34:57
%S -3,-4,0,-4,0,16,0,20,18,24,0,-10,0,32,28,100,0,148,0,198,403,48,0,82,
%T 250,56,18,138,0,752,0,644,436,72,705,950,0,80,369,1178,0,1468,0,1322,
%U 448,96,0,1930,1029,1104,766,146,0,2488,1680,478,3058,120,0,2674,0
%N (Binomial(2n, n) - binomial(2n - 2, n - 1)) (mod n^2) - n - 2.
%C Conjecture: a(n) = 0 iff n is an odd prime.
%C a(n) < 0 if n = 1, 2, 4, 12, 924, 1287, 2002, 2145, 3366, 3640, ... .
%C a(n) is odd if n = 1, 21, 35, 39, 49, 63, 69, 85, 91, 119, 123, ... .
%H Vincenzo Librandi, <a href="/A220956/b220956.txt">Table of n, a(n) for n = 1..10100</a>
%F a(n) = A051924(n) (mod n^2) -n -2.
%e a(8)=20 since C(16,8) - C(14,7) (mod 64) = (12870 - 3432) (mod 64) = 9438 (mod 64) = 30 and 30 -8 -2 = 20.
%t f[n_] := Mod[Binomial[2 n, n] - Binomial[2 n - 2, n - 1], n^2] - n - 2; Array[f, 61]
%o (Magma) [(Binomial(2*n,n)-Binomial(2*n-2,n-1)) mod n^2-n-2: n in [1..70]]; // _Bruno Berselli_, Feb 21 2013
%Y Cf. A051924, A000984.
%K sign,easy
%O 1,1
%A _Gary Detlefs_ and _Robert G. Wilson v_, Feb 20 2013
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