A140230 Binomial transform of [1, 2, -3, -4, 5, 6, -7, -8, 9, 10, ...].

1, 3, 2, -6, -20, -28, -8, 56, 144, 176, 32, -352, -832, -960, -128, 1920, 4352, 4864, 512, -9728, -21504, -23552, -2048, 47104, 102400, 110592, 8192, -221184, -475136, -507904, -32768, 1015808, 2162688, 2293760, 131072, -4587520, -9699328

Offset: 1

Author

Gary W. Adamson, May 13 2008

Keywords

• sign
• easy

Examples

			a(4) = -6 = (1, 3, 3, 1) dot (1, 2, -3, -4) = (1 + 6 - 9 - 4).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..206
• Index entries for linear recurrences with constant coefficients, signature (4,-8,8,-4).

Programs

PARI

Vec((2*x^3-2*x^2-x+1)/(4*x^4-8*x^3+8*x^2-4*x+1)+O(x^66)) /* Joerg Arndt, Apr 25 2011 */

Formula

A007318 * [1, 2, -3, -4, 5, 6, -7, -8, 9, 10, ...]; i.e., (+) signs when n == 1 or 2 (mod 4); (-) otherwise.

a(1+4*n) + a(2+4*n) + a(3+4*n) + a(4+4*n) = 0. - Paul Curtz, Apr 22 2011

a(n) = 4*a(n-1) - 8*a(n-2) + 8*a(n-3) - 4*a(n-4). - Joerg Arndt, Apr 25 2011

G.f.: x*(x-1)*(2*x^2-1) / (1-2*x+2*x^2)^2. - R. J. Mathar, Jun 02 2011

a(n) = Sum_{k=0..n-1} binomial(n-1,k)*(n-k)*(-1)^floor((n-k-1)/2). - Wesley Ivan Hurt, Sep 04 2022