A370173 Riordan array (1-x-x^2, x*(1+x)).

1, -1, 1, -1, 0, 1, 0, -2, 1, 1, 0, -1, -2, 2, 1, 0, 0, -3, -1, 3, 1, 0, 0, -1, -5, 1, 4, 1, 0, 0, 0, -4, -6, 4, 5, 1, 0, 0, 0, -1, -9, -5, 8, 6, 1, 0, 0, 0, 0, -5, -15, -1, 13, 7, 1, 0, 0, 0, 0, -1, -14, -20, 7, 19, 8, 1, 0, 0, 0, 0, 0, -6, -29, -21, 20, 26, 9, 1

Offset: 0

Author

Philippe Deléham, Feb 27 2024

Keywords

• sign
• tabl
• easy

Comments

Triangle T(n,k) read by rows : matrix product of A155112*A130595.

Triangle T(n,k), read by rows, given by [-1, 2, -1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Examples

			Triangle T(n,k) begins:
  1;
 -1,  1;
 -1,  0,  1;
  0, -2,  1,  1;
  0, -1, -2,  2, 1;
  0,  0, -3, -1, 3, 1;
...
		

Crossrefs

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.

Cf. A084938, A130595, A155112.

Programs

Python

from functools import cache
@cache
def T(n, k):
    if k > n: return 0
    if n == 0: return 1
    if k == 0: return -1 if n == 1 or n == 2 else 0
    return T(n-1, k-1) + T(n-2, k-1)
for n in range(9):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Feb 28 2024

Formula

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

T(n,k) = Sum_{j = k..n} A155112(n,j)*A130595(j,k).

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.