A344566 T(n, k) = (-1)^(n - k)*binomial(n - 1, k - 1)*hypergeom([-(n - k)/2, -(n - k - 1)/2], [1 - n], 4). Triangle read by rows, T(n, k) for 0 <= k <= n.

1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 1, 1, -3, 1, 0, -1, 2, 3, -4, 1, 0, 0, -4, 2, 6, -5, 1, 0, 1, 2, -9, 0, 10, -6, 1, 0, -1, 3, 9, -15, -5, 15, -7, 1, 0, 0, -6, 3, 24, -20, -14, 21, -8, 1, 0, 1, 3, -18, -6, 49, -21, -28, 28, -9, 1

Offset: 0

Author

Peter Luschny, May 23 2021

Keywords

• sign
• tabl

Comments

The inverse of the Riordan array for directed animals A122896. Without the first column (1, 0, 0, ...) the inverse of the Motzkin triangle A064189.

Examples

			Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0, -1,  1;
[3] 0,  0, -2,  1;
[4] 0,  1,  1, -3,   1;
[5] 0, -1,  2,  3,  -4,   1;
[6] 0,  0, -4,  2,   6,  -5,   1;
[7] 0,  1,  2, -9,   0,  10,  -6, 1;
[8] 0, -1,  3,  9, -15,  -5,  15, -7,  1;
[9] 0,  0, -6,  3,  24, -20, -14, 21, -8, 1.
		

Crossrefs

A117569 (row sums).

Cf. A122896, A064189.

Programs

Maple

T := (n,k) -> (-1)^(n-k)*binomial(n-1,k-1)*hypergeom([-(n-k)/2, -(n-k-1)/2], [1-n], 4): seq(seq(simplify(T(n, k)), k=0..n), n = 0..10);

SageMath

# uses[riordan_array from A256893]
riordan_array(1, x / (1 + x + x^2), 10)

Formula

Riordan_array (1, x / (1 + x + x^2)).