A244213 Inverse binomial transform of -2 followed by A000032(n+1).

-2, 3, -1, 0, 3, -7, 14, -25, 43, -72, 119, -195, 318, -517, 839, -1360, 2203, -3567, 5774, -9345, 15123, -24472, 39599, -64075, 103678, -167757, 271439, -439200, 710643, -1149847, 1860494, -3010345, 4870843, -7881192

Offset: 0

Author

Paul Curtz, Jun 23 2014

Keywords

• sign
• easy

Comments

A simple transform of a(n) is a(n) with -a(0) instead of nonzero a(0) (or -a(0) followed by a(n+1)). Example: -1 followed by A198631(n+1)/A006519(n+2). Its inverse binomial transform is -1, 3/2, -2, 9/4, -2, 3/2, -2,... = -(-1)^n*A143074(n).

Difference table of -2 followed by A000032(n+1):

-2, 1, 3, 4, 7, 11, 18,...

3, 2, 1, 3, 4, 7, 11,...

-1, -1, 2, 1, 3, 4, 7,...

0, 3, -1, 2, 1, 3, 4,...

3, -4, 3, -1, 2, 1, 3,...

-7, 7, -4, 3, -1, 2, 1,...

14, -11, 7, -4, 3, -1, 2,...

etc.

a(n) is the first column.

Links

• Index entries for linear recurrences with constant coefficients, signature (-2,0,1).

Crossrefs

Cf. A000045, A000032, A000204.

Programs

PARI

Vec(-(5*x^2-x-2)/((x+1)*(x^2-x-1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

Formula

a(n) = -2, 3, -1, followed by -(-1)^n*A206417(n).

a(n) = (-1)^n* (A000032(n) - 4).

a(n+3) = -a(n) -(-1)^n*A022112(n).

a(n) = -2*a(n-1) + a(n-3). - Colin Barker, Jun 23 2014

G.f.: -(5*x^2-x-2) / ((x+1)*(x^2-x-1)). - Colin Barker, Jun 23 2014