A210860 Triangle of coefficients of polynomials u(n,x) jointly generated with A210861; see the Formula section of A210861.

1, 2, 1, 4, 5, 2, 10, 18, 12, 3, 26, 64, 62, 28, 5, 76, 230, 286, 183, 60, 8, 232, 846, 1298, 1073, 503, 126, 13, 764, 3220, 5832, 5884, 3563, 1288, 255, 21, 2620, 12608, 26436, 31530, 23353, 10956, 3158, 506, 34, 9496, 51084, 121276, 166630

Offset: 1

Author

Clark Kimberling, Mar 28 2012

Keywords

• nonn
• tabl

Comments

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

Column 1: A000085

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

Examples

			First five rows:
   1
   2    1
   4    5    2
  10   18   12    3
  26   64   62   28    5
First three polynomials u(n,x): 1, 2 + x, 4 + 5x + 2x^2.
		

Crossrefs

Cf. A210861, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + (x + 1)*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[%]   (* A210860 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210861 *)

Formula

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

Extensions

Definition clarified by Alonso del Arte and Harvey P. Dale, Dec 17 2012