A091005 Expansion of x^2/((1-2*x)*(1+3*x)).

0, 0, 1, -1, 7, -13, 55, -133, 463, -1261, 4039, -11605, 35839, -105469, 320503, -953317, 2876335, -8596237, 25854247, -77431669, 232557151, -697147165, 2092490071, -6275373061, 18830313487, -56482551853, 169464432775, -508359743893, 1525146340543

Offset

0,5

Comments

Inverse binomial transform of A091002.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-1,6).

Formula

2^n = A091003(n) + 3*A091004(n) + 6*a(n).

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

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

Mathematica

a[n_]:=(MatrixPower[{{1, 4}, {1, -2}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Join[{0, 0}, LinearRecurrence[{-1, 6}, {1, -1}, 30]] (* G. C. Greubel, Feb 01 2019 *)
CoefficientList[Series[x^2/((1-2x)(1+3x)), {x, 0, 30}], x] (* Harvey P. Dale, Apr 30 2022 *)

Prog

(PARI) vector(30, n, n--; (3*2^n + 2*(-3)^n - 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019
(Magma) [0] cat [(3*2^n + 2*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019
(Sage) [0] + [(3*2^n + 2*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019
(GAP) Concatenation([0], List([1..30], n -> (3*2^n + 2*(-3)^n)/30)) # G. C. Greubel, Feb 01 2019

Crossrefs

Cf. A015441.

Sequence in context: A018562 A112540 A193489 * A015441 A255286 A253210

Adjacent sequences: A091002 A091003 A091004 * A091006 A091007 A091008

Keyword

easy,sign

Author

Paul Barry, Dec 13 2003

Status

approved