A122918 Expansion of (1+x)^2/(1+x+x^2)^2.

1, 0, -2, 2, 1, -4, 3, 2, -6, 4, 3, -8, 5, 4, -10, 6, 5, -12, 7, 6, -14, 8, 7, -16, 9, 8, -18, 10, 9, -20, 11, 10, -22, 12, 11, -24, 13, 12, -26, 14, 13, -28, 15, 14, -30, 16, 15, -32, 17, 16, -34, 18, 17, -36, 19, 18, -38

Offset: 0

Author

Paul Barry, Sep 19 2006

Keywords

• easy
• sign

Comments

Row sums of Riordan array (1/(1+x+x^2), x/(1+x)^2), A122917.

For n>=1, a(n) equals (-1)^(n+1) times the second immanant of the n X n matrix with 1's along the main diagonal, superdiagonal, and subdiagonal, and 0's everywhere else. The second immanant of an n X n matrix A is the immanant of A given by the partition (2, 1^(n-2)). - John M. Campbell, Apr 12 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-2,-3,-2,-1).

Crossrefs

Cf. A187430 (series reversion, with offset 1).

Programs

Mathematica

CoefficientList[Series[(1 + x)^2/(1 + x + x^2)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Apr 13 2014 *)
Print[Table[(-1)^(n+1)*Sum[Binomial[n-i, i]*(n-2*i-1)*(-1)^i, {i, 0, Floor[n/2]}], {n, 0, 100}]] ;  (* John M. Campbell, Jan 08 2016 *)

PARI

Vec((1+x)^2/(1+x+x^2)^2 + O(x^100)) \\ Altug Alkan, Jan 08 2015

PARI

A122918(n)=(-1)^(n+1)*sum(i=0,n\2,(-1)^i*binomial(n-i,i)*(n-2*i-1)) \\ M. F. Hasler, Jan 12 2016

Formula

a(n) = 4 * sqrt(3) * cos(2*Pi*n/3 + Pi/6)/9 + 2(n+1) * sin(2*Pi*n/3 + Pi/6)/3. a(n) = sum{k=0..n} A057078(k) * A057078(n-k).

a(n) = (-1)^(n+1)*sum((-1)^i*binomial(n-i,i)*(n-2*i-1), i=0..[n/2]). - John M. Campbell, Jan 08 2016