%I #14 Feb 18 2020 01:25:57
%S 1,1,1,-1,1,2,-1,-2,2,3,1,-2,-5,3,5,1,3,-5,-10,5,8,-1,3,9,-10,-20,8,
%T 13,-1,-4,9,22,-20,-38,13,21,1,-4,-14,22,51,-38,-71,21,34,1,5,-14,-40,
%U 51,111,-71,-130,34,55
%N Triangle T(n,k), read by rows, given by (1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C T(n,n) = A000045(n+1) = Fibonacci(n+1).
%F T(n,k) = T(n-1,k-1) + T(n-2,k-2) - T(n-2,k) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n<k.
%F G.f.: (1+x)/(1-y*x+(1-y^2)*x^2).
%F Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A046717(n), A000007(n), A057077(n), A040000(n), A000244(n) for x = -2, -1, 0, 1, 2 respectively.
%e Triangle begins :
%e 1
%e 1, 1
%e -1, 1, 2
%e -1, -2, 2, 3
%e 1, -2, -5, 3, 5
%e 1, 3, -5, -10, 5, 8
%e -1, 3, 9, -10, -20, 8, 13
%t With[{m = 9}, CoefficientList[CoefficientList[Series[(1+x)/(1-y*x+(1-y^2)*x
%t ^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* _Georg Fischer_, Feb 17 2020 *)
%o (PARI) T(n,k) = if (k<0, 0, if (n<k, 0, if ((k<=1) && (n<=1), 1, T(n-1,k-1) + T(n-2,k-2) - T(n-2,k))));
%o matrix(10, 10, n, k, T(n-1,k-1)) \\ to see the triangle \\ _Michel Marcus_, Feb 17 2020
%Y Cf. A000007, A000045, A000244, A040000, A046717, A057077, A084938, A124137, A123585.
%K sign,tabl
%O 0,6
%A _Philippe Deléham_, Dec 18 2011
%E a(52) corrected by _Georg Fischer_, Feb 17 2020