login
A208756
Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section.
4
1, 0, 2, 0, 1, 4, 0, 1, 3, 8, 0, 1, 3, 9, 16, 0, 1, 3, 11, 23, 32, 0, 1, 3, 13, 31, 57, 64, 0, 1, 3, 15, 39, 87, 135, 128, 0, 1, 3, 17, 47, 121, 227, 313, 256, 0, 1, 3, 19, 55, 159, 339, 579, 711, 512, 0, 1, 3, 21, 63, 201, 471, 933, 1431, 1593, 1024, 0, 1, 3, 23, 71
OFFSET
1,3
COMMENTS
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0<=k<=n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012
FORMULA
u(n,x)=u(n-1,x)+2x*v(n-1,x),
v(n,x)=x*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As triangle with 0<=k<=n : G.f.: (1-x+y*x)/(1-(1+y)*x-(2*y^2-y)*x^2). - Philippe Deléham, Mar 02 2012
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 02 2012
EXAMPLE
First five rows:
1
0...2
0...1...4
0...1...3...8
0...1...3...9...16
First five polynomials v(n,x):
1
2x
x + 4x^2
x + 3x^2 + 8x^3
x + 3x^2 + 9x^3 + 16^4
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 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[%] (* A208755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208756 *)
CROSSREFS
Sequence in context: A372873 A212206 A247489 * A259873 A121462 A349706
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 01 2012
STATUS
approved