OFFSET
1,3
COMMENTS
FORMULA
u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 29 2012
EXAMPLE
First five rows:
1
1...2
3...3....3
3...11...8....5
9...18...29...17...8
First three polynomials v(n,x): 1, 1 + 2x, 3 + 3x + 3x^2
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 = 1; c = 0; h = 2; p = -1; 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[%] (* A210793 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210794 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 26 2012
STATUS
approved