A316386 Binomial transform of [0, 1, 2, -3, -4, 5, 6, -7, -8, ...].

0, 1, 4, 6, 0, -20, -48, -56, 0, 144, 320, 352, 0, -832, -1792, -1920, 0, 4352, 9216, 9728, 0, -21504, -45056, -47104, 0, 102400, 212992, 221184, 0, -475136, -983040, -1015808, 0, 2162688, 4456448, 4587520, 0, -9699328, -19922944, -20447232, 0, 42991616

Offset: 0

Author

Paul Curtz, Jul 01 2018

Keywords

• sign
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-8,8,-4).

Crossrefs

Cf. A009545, A140230.

Programs

Maple

seq(coeff(series(x*(1-2*x^2)/(1-2*x+2*x^2)^2, x,n+1),x,n),n=0..45); # Muniru A Asiru, Jul 01 2018

Mathematica

CoefficientList[Series[x (1 - 2 x^2)/(1 - 2 x + 2 x^2)^2, {x, 0, 41}], x] (* Michael De Vlieger, Jul 01 2018 *)
LinearRecurrence[{4, -8, 8, -4}, {0, 1, 4, 6}, 42] (* Robert G. Wilson v, Jul 15 2018 *)

PARI

concat(0, Vec(x*(1 - 2*x^2) / (1 - 2*x + 2*x^2)^2 + O(x^40))) \\ Colin Barker, Jul 01 2018

Formula

a(n) = n * A009545(n).

a(n+1) = a(n) + A140230(n).

From Colin Barker, Jul 01 2018: (Start)

G.f.: x*(1 - 2*x^2) / (1 - 2*x + 2*x^2)^2.

a(n) = 4*a(n-1) - 8*a(n-2) + 8*a(n-3) - 4*a(n-4) for n>3.

a(n) = i*(((1-i)^n - (1+i)^n)*n) / 2 where i=sqrt(-1).

(End)

Extensions

More terms from Colin Barker, Jul 01 2018