A127896 Expansion of 1/(1 + 2*x + 3*x^2 + x^3).

1, -2, 1, 3, -7, 4, 10, -25, 16, 33, -89, 63, 108, -316, 245, 350, -1119, 943, 1121, -3952, 3598, 3539, -13920, 13625, 10971, -48897, 51256, 33208, -171287, 191694, 97265, -598325, 713161, 271388, -2083934

Offset: 0

Author

Paul Barry, Feb 04 2007

Keywords

• easy
• sign

Comments

Row sums of A127895. Series reversion is A127897.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (-2,-3,-1).

Crossrefs

Cf. A127895, A127897.

Programs

Magma

I:=[1, -2, 1]; [n le 3 select I[n] else -2*Self(n-1) -3*Self(n-2) -Self(n-3): n in [1..50]]; // G. C. Greubel, Apr 29 2018

Mathematica

CoefficientList[Series[1/(1+2x+3x^2+x^3),{x,0,40}],x]  (* Harvey P. Dale, Apr 19 2011 *)
LinearRecurrence[{-2, -3, -1}, {1, -2, 1}, 30] (* G. C. Greubel, Apr 29 2018 *)

PARI

x='x+O('x^50); Vec(1/(1+2*x+3*x^2+x^3)) \\ G. C. Greubel, Apr 29 2018

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n+2k+2,n-k).

a(n) = -2*a(n-1) -3*a(n-2) -a(n-3), n>=3. - Vincenzo Librandi, Mar 22 2011