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

%I A370173 #14 Feb 29 2024 10:50:10
%S A370173 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,
%T A370173 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,
%U A370173 -1,-14,-20,7,19,8,1,0,0,0,0,0,-6,-29,-21,20,26,9,1
%N A370173 Riordan array (1-x-x^2, x*(1+x)).
%C A370173 Triangle T(n,k) read by rows : matrix product of A155112*A130595.
%C A370173 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.
%F A370173 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.
%F A370173 T(n,k) = Sum_{j = k..n} A155112(n,j)*A130595(j,k).
%F A370173 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.
%e A370173 Triangle T(n,k) begins:
%e A370173   1;
%e A370173  -1,  1;
%e A370173  -1,  0,  1;
%e A370173   0, -2,  1,  1;
%e A370173   0, -1, -2,  2, 1;
%e A370173   0,  0, -3, -1, 3, 1;
%e A370173 ...
%o A370173 (Python)
%o A370173 from functools import cache
%o A370173 @cache
%o A370173 def T(n, k):
%o A370173     if k > n: return 0
%o A370173     if n == 0: return 1
%o A370173     if k == 0: return -1 if n == 1 or n == 2 else 0
%o A370173     return T(n-1, k-1) + T(n-2, k-1)
%o A370173 for n in range(9):
%o A370173     print([T(n, k) for k in range(n+1)])  # _Peter Luschny_, Feb 28 2024
%Y A370173 Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
%Y A370173 Cf. A084938, A130595, A155112.
%K A370173 sign,tabl,easy
%O A370173 0,8
%A A370173 _Philippe Deléham_, Feb 27 2024