OFFSET
1,5
COMMENTS
Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012
FORMULA
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = A001045(k+1)*binomial(n-1,k). - Philippe Deléham, Mar 18 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012
EXAMPLE
First five rows:
1
1...1
1...2...3
1...3...9....5
1...4...18...20...11
First five polynomials u(n,x):
1, 1 + x, 1 + 2x + 3x^2, 1 + 3x + 9x^2 + 5x^3, 1 + 4x + 18x^2 + 20x^3 + 11x^4.
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 2, 3, 0
1, 3, 9, 5, 0
1, 4, 18, 20, 11, 0
1, 5, 30, 50, 55, 21, 0. - Philippe Deléham, Mar 18 2012
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
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[%] (* A208330 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208331 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved