OFFSET
1,3
COMMENTS
Row n starts with 1 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 2: A052944.
Row sums: A000244 (powers of 3).
Alternating row sums: A001333.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 1/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 29 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1 - 2*x - y*x + 3*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
EXAMPLE
First five rows:
1;
1, 2;
1, 5, 3;
1, 10, 11, 5;
1, 19, 28, 25, 8;
First three polynomials v(n,x):
1
1 + 2x
1 + 5x + 3x^2
From Philippe Deléham, Mar 29 2012: (Start)
(1, 0, 1/2, 3/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 5, 3, 0;
1, 10, 11, 5, 0;
1, 19, 28, 25, 8, 0;
1, 36, 62, 81, 50, 13, 0;
1, 69, 129, 218, 193, 98, 21, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = -1; p = 2; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A210791 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210792 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007051 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A001129 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A001333 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 26 2012
STATUS
approved