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

0,9

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.

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

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.

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);

Prog

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

Crossrefs

A117569 (row sums).

Cf. A122896, A064189.

Sequence in context: A137560 A201093 A131255 * A198295 A221857 A133607

Adjacent sequences: A344563 A344564 A344565 * A344567 A344568 A344569

Keyword

sign,tabl

Author

Peter Luschny, May 23 2021

Status

approved