A091004 Expansion of x*(1-x)/((1-2*x)*(1+3*x)).

0, 1, -2, 8, -20, 68, -188, 596, -1724, 5300, -15644, 47444, -141308, 425972, -1273820, 3829652, -11472572, 34450484, -103285916, 309988820, -929704316, 2789637236, -8367863132, 25105686548, -75312865340, 225946984628, -677824176668, 2033506084436

Offset: 0

Author

Paul Barry, Dec 13 2003

Keywords

• easy
• sign

Comments

Inverse binomial transform of A091001.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,6).

Crossrefs

Cf. A091001, A091003, A091005, A112555.

Programs

GAP

Concatenation([0], List([1..30], n -> (3*2^n - 8*(-3)^n)/30)); # G. C. Greubel, Feb 01 2019

Magma

[0] cat [(3*2^n - 8*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019

Mathematica

CoefficientList[Series[x(1-x)/((1-2x)(1+3x)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 17 2017 *)
Join[{0}, LinearRecurrence[{-1, 6}, {1, -2}, 30]] (* G. C. Greubel, Feb 01 2019 *)

PARI

vector(30, n, n--; (3*2^n - 8*(-3)^n + 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019

Sage

[0] + [(3*2^n - 8*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019

Formula

G.f.: x*(1-x)/((1-2*x)*(1+3*x)).

a(n) = (3*2^n - 8*(-3)^n + 5*0^n)/30.

2^n = A091003(n) + 3*a(n) + 6*A091005(n).

a(n+1) = Sum_{k=0..n} A112555(n,k)*(-3)^k. - Philippe Deléham, Sep 11 2009

E.g.f.: (3*exp(2*x) - 8*exp(-3*x) + 5)/30. - G. C. Greubel, Feb 01 2019