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A208333
Triangle of coefficients of polynomials v(n,x) jointly generated with A208332; see the Formula section.
3
1, 0, 4, 0, 2, 10, 0, 2, 6, 28, 0, 2, 6, 24, 76, 0, 2, 6, 28, 80, 208, 0, 2, 6, 32, 100, 264, 568, 0, 2, 6, 36, 120, 360, 840, 1552, 0, 2, 6, 40, 140, 464, 1232, 2624, 4240, 0, 2, 6, 44, 160, 576, 1680, 4128, 8064, 11584, 0, 2, 6, 48, 180, 696, 2184, 5952
OFFSET
1,3
COMMENTS
As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 28 2012
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 28 2012: (Start)
As triangle with 0 <= k <= n:
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 4 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-x+2*y*x)/(1-x-2*y*x+2*y*x^2-2*y^2*x^2).
T(n,n) = A026150(n+1).
Sum_{k=0..n} T(n,k) = A003946(n). (End)
EXAMPLE
First five rows:
1;
0, 4;
0, 2, 10;
0, 2, 6, 28;
0, 2, 6, 24, 76;
First five polynomials u(n,x):
1
4x
2x + 10x^2
2x + 6x^2 + 28x^3
2x + 6x^2 + 24x^3 + 76x^4.
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] + 2 x*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[%] (* A208332 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208333 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved