A208931 Triangle of coefficients of polynomials u(n,x) jointly generated with A208932; see the Formula section.

1, 1, 2, 1, 8, 4, 1, 18, 20, 8, 1, 32, 64, 56, 16, 1, 50, 160, 224, 136, 32, 1, 72, 340, 680, 664, 328, 64, 1, 98, 644, 1736, 2416, 1872, 760, 128, 1, 128, 1120, 3920, 7264, 7856, 4984, 1736, 256, 1, 162, 1824, 8064, 19056, 26992, 23768, 12832, 3896

Offset: 1

Author

Clark Kimberling, Mar 04 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
1
1...2
1...8....4
1...18...20...8
1...32...64...56...16
First five polynomials u(n,x):
1
1 + 2x
1 + 8x + 4x^2
1 + 18x + 20x^2 + 8x^3
1 + 32x + 64x^2 + 56x^3 + 16x^4
		

Crossrefs

Cf. A208931, A208510.

Programs

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 + 1)*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[%]    (* A208931 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208932 *)

Formula

u(n,x)=u(n-1,x)+2x*v(n-1,x),

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

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