A210858 Triangle of coefficients of polynomials u(n,x) jointly generated with A210859; see the Formula section.

1, 1, 1, 1, 3, 2, 1, 6, 8, 3, 1, 10, 24, 20, 5, 1, 15, 59, 82, 45, 8, 1, 21, 125, 271, 251, 98, 13, 1, 28, 237, 765, 1073, 701, 204, 21, 1, 36, 413, 1898, 3860, 3802, 1842, 414, 34, 1, 45, 674, 4235, 12115, 17011, 12325, 4589, 820, 55, 1, 55, 1044, 8662

Offset: 1

Author

Clark Kimberling, Mar 28 2012

Keywords

• nonn
• tabl

Comments

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).

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

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

Examples

			First five rows:
1
1...1
1...3....2
1...6....8....3
1...10...24...20...5
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + 2x^2.
		

Crossrefs

Cf. A210806, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x];
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[%]   (* A210858 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210859 *)

Formula

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

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

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