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A370173
Riordan array (1-x-x^2, x*(1+x)).
0
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
0,8
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.
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.
EXAMPLE
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;
...
PROG
(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
KEYWORD
sign,tabl,easy
AUTHOR
Philippe Deléham, Feb 27 2024
STATUS
approved