A209770 Triangle of coefficients of polynomials v(n,x) jointly generated with A209769; see the Formula section.

1, 3, 1, 5, 4, 2, 9, 12, 10, 3, 15, 29, 33, 19, 5, 25, 64, 93, 77, 37, 8, 41, 132, 234, 251, 171, 69, 13, 67, 261, 548, 719, 629, 362, 127, 21, 109, 500, 1216, 1884, 2004, 1482, 742, 230, 34, 177, 936, 2592, 4628, 5784, 5196, 3342, 1482, 412, 55, 287

Offset: 1

Author

Clark Kimberling, Mar 15 2012

Keywords

• nonn
• tabl

Comments

Column 1: A001595

Row n ends with F(n), where F=A000045, the Fibonacci numbers.

Row sums: 1,4,11,34,101,304,911,2734,... A060925

Alternating row sums: 1,2,3,4,5,6,7,.... A000027

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

Examples

			First five rows:
1
3....1
5....4....2
9....12...10...3
15...29...33...19...5
First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 2x^2.
		

Crossrefs

Cf. A209669, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*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[%]    (* A209769 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209770 *)

Formula

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