A209687 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section.

1, 0, 2, 0, 1, 5, 0, 1, 5, 12, 0, 1, 6, 18, 29, 0, 1, 7, 26, 58, 70, 0, 1, 8, 35, 98, 175, 169, 0, 1, 9, 45, 149, 339, 507, 408, 0, 1, 10, 56, 212, 574, 1108, 1428, 985, 0, 1, 11, 68, 288, 894, 2066, 3476, 3940, 2378, 0, 1, 12, 81, 378, 1314, 3492, 7074, 10572

Offset: 1

Author

Clark Kimberling, Mar 12 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums: signed powers of 2. 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, ...) DELTA (2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 28 2012

Examples

			First five rows:
1
0...2
0...1...5
0...1...5...12
0...1...6...18...29
First three polynomials v(n,x): 1, 2x, x + 5x^2.
		

Crossrefs

Cf. A208339, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 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[%]   (* A209687 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A208339 *)

Formula

u(n,x)=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) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 28 2012

G.f.: (-1+x)/(-1+x+2*x*y-x^2*y+x^2*y^2). - R. J. Mathar, Aug 12 2015