A220956 (Binomial(2n, n) - binomial(2n - 2, n - 1)) (mod n^2) - n - 2.

-3, -4, 0, -4, 0, 16, 0, 20, 18, 24, 0, -10, 0, 32, 28, 100, 0, 148, 0, 198, 403, 48, 0, 82, 250, 56, 18, 138, 0, 752, 0, 644, 436, 72, 705, 950, 0, 80, 369, 1178, 0, 1468, 0, 1322, 448, 96, 0, 1930, 1029, 1104, 766, 146, 0, 2488, 1680, 478, 3058, 120, 0, 2674, 0

Offset: 1

Author

Gary Detlefs and Robert G. Wilson v, Feb 20 2013

Keywords

• sign
• easy

Comments

Conjecture: a(n) = 0 iff n is an odd prime.

a(n) < 0 if n = 1, 2, 4, 12, 924, 1287, 2002, 2145, 3366, 3640, ... .

a(n) is odd if n = 1, 21, 35, 39, 49, 63, 69, 85, 91, 119, 123, ... .

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10100

Crossrefs

Cf. A051924, A000984.

Programs

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

Mathematica

f[n_] := Mod[Binomial[2 n, n] - Binomial[2 n - 2, n - 1], n^2] - n - 2; Array[f, 61]

Formula

a(n) = A051924(n) (mod n^2) -n -2.