A209703 Triangle of coefficients of polynomials u(n,x) jointly generated with A209704; see the Formula section.

1, 0, 2, 0, 3, 3, 0, 4, 6, 5, 0, 5, 10, 14, 8, 0, 6, 15, 28, 28, 13, 0, 7, 21, 48, 66, 55, 21, 0, 8, 28, 75, 129, 149, 104, 34, 0, 9, 36, 110, 225, 326, 319, 193, 55, 0, 10, 45, 154, 363, 626, 774, 661, 352, 89, 0, 11, 55, 208, 553, 1099, 1625, 1761, 1332, 634

Offset: 1

Author

Clark Kimberling, Mar 12 2012

Keywords

• nonn
• tabl

Comments

For n>1, row n begins with 0, has second term n, and ends with F(n+1), where F=A000045 (Fibonacci numbers); for n>2, column 2 consists of triangular numbers.

Row sums: A098790.

Alternating row sums: 1,-2,0,-3,-1,-4,-3,-5,-3,-6,,...

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

Examples

			First five rows:
1
0...2
0...3....3
0...4....6...5
0...5...10...14...8
First three polynomials v(n,x): 1, 2x, 3x + 3x^2.
		

Crossrefs

Cf. A209704, 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_] := (x + 1)*u[n - 1, x] + 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[%]  (* A209703 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A209704 *)

Formula

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

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

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