A208459 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.

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

Offset

0,15

Comments

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

Links

Table of n, a(n) for n=0..65.

Formula

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.

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

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.

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.

Example

Triangle begins :
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, -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, 18, -16, 15, 3, -25, 34

Crossrefs

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

Sequence in context: A053473 A345388 A322983 * A144764 A084929 A293575

Adjacent sequences: A208456 A208457 A208458 * A208460 A208461 A208462

Keyword

easy,sign,tabl

Author

Philippe Deléham, Feb 27 2012

Status

approved