A118409 -id:A118409 - cp's OEIS frontend

A118407 Triangle, read by rows, equal to the matrix square of triangle A118404; also equals the matrix inverse of triangle A118401.

1, 0, 1, -2, 0, 1, 2, -2, 0, 1, 0, 2, -2, 0, 1, -2, 0, 2, -2, 0, 1, 4, -2, 0, 2, -2, 0, 1, -6, 4, -2, 0, 2, -2, 0, 1, 4, -6, 4, -2, 0, 2, -2, 0, 1, 6, 4, -6, 4, -2, 0, 2, -2, 0, 1, -20, 6, 4, -6, 4, -2, 0, 2, -2, 0, 1, 26, -20, 6, 4, -6, 4, -2, 0, 2, -2, 0, 1, -12, 26, -20, 6, 4, -6, 4, -2, 0, 2, -2, 0, 1

Offset: 0

Paul D. Hanna, Apr 27 2006

This triangle has an integer matrix square-root (A118404) if the main diagonal of the square-root is allowed to be signed. Even though the columns of this triangle are all the same, the columns of the matrix square-root A118404 are all different.

			Triangle begins:
1;
0, 1;
-2, 0, 1;
2,-2, 0, 1;
0, 2,-2, 0, 1;
-2, 0, 2,-2, 0, 1;
4,-2, 0, 2,-2, 0, 1;
-6, 4,-2, 0, 2,-2, 0, 1;
4,-6, 4,-2, 0, 2,-2, 0, 1;
6, 4,-6, 4,-2, 0, 2,-2, 0, 1;
-20, 6, 4,-6, 4,-2, 0, 2,-2, 0, 1;
26,-20, 6, 4,-6, 4,-2, 0, 2,-2, 0, 1; ...

Cf. A118404 (matrix square-root), A118401 (matrix inverse), A118408 (row sums), A118409 (unsigned row sums).

{T(n,k)=polcoeff(polcoeff((1+x)^2/(1+x^2)/(1+2*x+2*x^2)/(1-x*y+x*O(x^n)),n,x)+y*O(y^k),k,y)}

G.f.: A(x,y) = (1+x)^2/(1+x^2)/(1+2*x+2*x^2)/(1-x*y). Column g.f.: (1+x)^2/(1+x^2)/(1+2*x+2*x^2).

A118408 Row sums of triangle A118407.

Original entry on oeis.org

1, 1, -1, 1, 1, -1, 3, -3, 1, 7, -13, 13, 1, -25, 51, -51, 1, 103, -205, 205, 1, -409, 819, -819, 1, 1639, -3277, 3277, 1, -6553, 13107, -13107, 1, 26215, -52429, 52429, 1, -104857, 209715, -209715, 1, 419431, -838861, 838861, 1, -1677721, 3355443, -3355443, 1, 6710887, -13421773, 13421773, 1

Offset: 0

Paul D. Hanna, Apr 27 2006

sign

Index entries for linear recurrences with constant coefficients, signature (-1, -1, 1, 0, 2).

Cf. A118407 (triangle), A118409 (unsigned row sums).

{a(n)=polcoeff((1+x)^2/(1+x^2)/(1+2*x+2*x^2)/(1-x+x*O(x^n)),n)}

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

cp's OEIS Frontend

A118407 Triangle, read by rows, equal to the matrix square of triangle A118404; also equals the matrix inverse of triangle A118401.

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