reviewed

approved

proposed

reviewed

editing

proposed

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938 . - Philippe Deléham, Apr 09 2012

u(n,x) = u(n-1,x) + x*v(n-1,x),

v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),

Contribution from _From _Philippe Deléham_, Apr 09 2012. : (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n :

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

1;

1..., 1;

1..., 2..., 3;

1..., 3..., 7...., 7;

1..., 4..., 12..., 20..., 17;

1

1 + x

1 + 2x + 3x^2

1 + 3x + 7x^2 + 7x^3

1 + 4x + 12x^2 + 20x^3 + 17x^4

From Philippe Deléham, Apr 09 2012: (Start)

(1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins :

1;

1, 0;

1, 1, 0;

1, 2, 3, 0;

1, 3, 7, 7, 0;

1, 4, 12, 20, 17, 0;

1, 5, 18, 40, 57, 41, 0 . - _Philippe Deléham_, Apr 09 2012; (End)

approved

editing

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938 . - _Philippe Deléham, _, Apr 09 2012

Contribution from _Philippe Deléham, _, Apr 09 2012. (Start)

1, 5, 18, 40, 57, 41, 0 . - _Philippe Deléham, _, Apr 09 2012

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938 . - DELEHAM Philippe, Deléham, Apr 09 2012

Contribution from DELEHAM Philippe, Deléham, Apr 09 2012. (Start)

1, 5, 18, 40, 57, 41, 0 . - DELEHAM Philippe, Deléham, Apr 09 2012

proposed

approved

editing

proposed

Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938 . - DELEHAM Philippe, Apr 09 2012

Contribution from DELEHAM Philippe, Apr 09 2012. (Start)

As DELTA-triangle T(n,k) with 0<=k<=n :

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)

(1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins :

1

1, 0

1, 1, 0

1, 2, 3, 0

1, 3, 7, 7, 0

1, 4, 12, 20, 17, 0

1, 5, 18, 40, 57, 41, 0 . - DELEHAM Philippe, Apr 09 2012

approved

editing

_Clark Kimberling (ck6(AT)evansville.edu), _, Feb 27 2012