A122917 -id:A122917 - cp's OEIS frontend

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

Paul Barry, Sep 19 2006

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

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

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

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 *)

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

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

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

A122919 Inverse of Riordan array (1/(1+x+x^2),x/(1+x)^2).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 10, 5, 1, 28, 34, 21, 7, 1, 90, 117, 83, 36, 9, 1, 297, 407, 319, 164, 55, 11, 1, 1001, 1430, 1209, 702, 285, 78, 13, 1, 3432, 5070, 4550, 2898, 1350, 454, 105, 15, 1, 11934, 18122, 17068, 11696, 6052

Offset: 0

Paul Barry, Sep 19 2006

Row sums are A097613. Diagonal sums are A122920. Inverse of A122917.

			Triangle begins
1,
1, 1,
3, 3, 1,
9, 10, 5, 1,
28, 34, 21, 7, 1,
90, 117, 83, 36, 9, 1,
297, 407, 319, 164, 55, 11, 1,
1001, 1430, 1209, 702, 285, 78, 13, 1

Riordan array ((1-3x+2x^2-(1-x)sqrt(1-4x))/(2x^2),(1-2x-sqrt(1-4x))/(2x))

cp's OEIS Frontend

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

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