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

1, 3, 1, 4, 3, 2, 5, 6, 8, 3, 6, 10, 18, 14, 5, 7, 15, 33, 38, 27, 8, 8, 21, 54, 81, 83, 49, 13, 9, 28, 82, 150, 197, 170, 89, 21, 10, 36, 118, 253, 401, 448, 342, 159, 34, 11, 45, 163, 399, 736, 999, 987, 671, 282, 55, 12, 55, 218, 598, 1253, 1988, 2387, 2106

Offset: 1

Author

Clark Kimberling, Mar 12 2012

Keywords

• nonn
• tabl

Comments

For n>1, row n starts with n+1, followed by the n-th

triangular number, and ends in F(n+1), where F=A000045

(Fibonacci numbers).

Column 3: A166830.

Row sums: A048654.

Alternating row sums: 1,2,3,4,5,6,7,8,9,...

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

Examples

			First five rows:
1
3...1
4...3....2
5...6....8....3
6...10...18...14...5
First three polynomials v(n,x): 1, 3 + x , 4 + 3x + 2x^2.
		

Crossrefs

Cf. A209703, 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.