login
Triangle T_x = T(n,k) given by (0, 1/x, 1-1/x, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (x, 1/x-1, -1/x, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938, for x = 0.
0

%I #6 Feb 22 2013 14:40:29

%S 1,0,0,0,1,1,0,1,0,-1,0,1,0,1,2,0,1,0,2,0,-3,0,1,0,3,-1,0,5,0,1,0,4,

%T -2,3,2,-8,0,1,0,5,-3,7,-2,-5,13,0,1,0,6,-4,12,-8,2,12,-21,0,1,0,7,-5,

%U 18,-16,15,3,-25,34

%N Triangle T_x = T(n,k) given by (0, 1/x, 1-1/x, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (x, 1/x-1, -1/x, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938, for x = 0.

%C Triangle T_x : T_1 = A103631, T_2 = A208343, T_3 = A208345.

%F T(n,k) = T(n-1,k) - T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) with T(0,0) = 1 T(1,0) = 0, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.

%F G.f.: (1-x+y*x)/(1-x+y*x- y^2*x^2-y*x^2).

%F Sum_{k, 0<=k<=n} T(n,k)*x^k = 12*A015548(n-1), 6*A085939(n-1), A106434(n), A000007(n), A000007(n), A077957(n), (-1)^n*A102901(n) for x = -4, -3, -2, -1, 0, 1, 2 respectively.

%F Sm_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000007(n), A034834(n-1), A077957(n), A052533(n), (-1)^n*A086344(n) for x = -1, 0, 1, 2, 3 respectively.

%e Triangle begins :

%e 1

%e 0, 0

%e 0, 1, 1

%e 0, 1, 0, -1

%e 0, 1, 0, 1, 2

%e 0, 1, 0, 2, 0, -3

%e 0, 1, 0, 3, -1, 0, 5

%e 0, 1, 0, 4, -2, 3, 2, -8

%e 0, 1, 0, 5, -3, 7, -2, -5, 13

%e 0, 1, 0, 6, -4, 12, -8, 2, 12, -21

%e 0, 1, 0, 7, -5, 18, -16, 15, 3, -25, 34

%Y Cf. A103631, A208343, A208345, A000045 (Fibonacci)

%K easy,sign,tabl

%O 0,15

%A _Philippe Deléham_, Feb 27 2012