A209693 Triangle of coefficients of polynomials u(n,x) jointly generated with A209694; see the Formula section.

1, 0, 2, 0, 3, 3, 0, 4, 9, 4, 0, 5, 17, 19, 5, 0, 6, 27, 49, 34, 6, 0, 7, 39, 98, 115, 55, 7, 0, 8, 53, 170, 284, 236, 83, 8, 0, 9, 69, 269, 585, 706, 440, 119, 9, 0, 10, 87, 399, 1070, 1706, 1568, 763, 164, 10, 0, 11, 107, 564, 1799, 3577, 4395, 3193, 1250, 219

Offset: 1

Author

Clark Kimberling, Mar 12 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums are periodic.

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
0...2
0...2...4
0...1...8...8
0...1...5...24...16
First three polynomials v(n,x): 1, 2x, 2x + 4x^2.
		

Crossrefs

Cf. A209694, 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_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]   (* A209693 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209694 *)

Formula

u(n,x)=x*u(n-1,x)+x*v(n-1,x),

v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.