A206474 Riordan array ((1+x-x^2)/(1-x^2), x/(1-x^2)).

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 0, 3, 3, 4, 4, 1, 1, 1, 1, 6, 6, 5, 5, 1, 1, 0, 4, 4, 10, 10, 6, 6, 1, 1, 1, 1, 10, 10, 15, 15, 7, 7, 1, 1, 0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1, 1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1

Offset: 0

Author

Philippe Deléham, Feb 08 2012

Keywords

• easy
• nonn
• tabl

Comments

Triangle T(n,k), read by rows, given by (1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Antidiagonal sums are A158780(n+1).

Row sums are 2*Fibonacci(n) = 2*A000045(n), n>0.

Examples

			Triangle begins :
1
1, 1
0, 1, 1
1, 1, 1, 1
0, 2, 2, 1, 1
1, 1, 3, 3, 1, 1
0, 3, 3, 4, 4, 1, 1
1, 1, 6, 6, 5, 5, 1, 1
0, 4, 4, 10, 10, 6, 6, 1, 1
1, 1, 10, 10, 15, 15, 7, 7, 1, 1
0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1
1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1
		

Crossrefs

Cf. A007318, A078812, A085478, A128908,

Programs

Mathematica

t[1, 0] = 1; t[2, 0] = 0; t[n_, n_] = 1; t[n_ /; n >= 0, k_ /; k >= 0] /; k <= n := t[n, k] = t[n-1, k-1] + t[n-2, k]; t[n_, k_] = 0; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 28 2013 *)

Formula

T(2n, 2k) = A128908(n,k), T(2n+1, 2k) = T(2n+1, 2k+1) = A085478(n,k) = Binomial (n+k, 2k), T(2n+2, 2k+1) = A078812(n,k) = Binomial(n+k-1, 2k-1).

T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(0,1) = 1, T(0,2) = 0.

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n* A000129(n) (n>0), A000007(n), A135528(n-1), A055389(n) for x = -2, -1, 0, 1 respectively .