A209567 Triangle of coefficients of polynomials u(n,x) jointly generated with A209568; see the Formula section.

1, 1, 1, 2, 3, 1, 3, 7, 6, 1, 4, 13, 18, 10, 1, 5, 21, 41, 39, 15, 1, 6, 31, 79, 108, 75, 21, 1, 7, 43, 136, 245, 250, 132, 28, 1, 8, 57, 216, 486, 661, 524, 217, 36, 1, 9, 73, 323, 875, 1497, 1601, 1015, 338, 45, 1, 10, 91, 461, 1464, 3031, 4109, 3556, 1844

Offset: 1

Author

Clark Kimberling, Mar 10 2012

Keywords

• nonn
• tabl

Comments

For n>1, row n begins with n and ends with 1.

For n>1, penultimate number in row n is (n-1)st triangular number.

Alternating row sums: 1,0,0,1,0,0,1,0,0,1,0,0,...

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

Examples

			First five rows:
1
1...1
2...3....1
3...7....6....1
4...13...18...10...1
First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.
		

Crossrefs

Cf. A209568, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 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[%]   (* A209567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209568 *)

Formula

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

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

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