A210866 Triangle of coefficients of polynomials u(n,x) jointly generated with A210867; see the Formula section.

1, 1, 1, 1, 3, 1, 1, 6, 6, 2, 1, 10, 21, 14, 3, 1, 15, 55, 65, 31, 5, 1, 21, 120, 235, 187, 65, 8, 1, 28, 231, 700, 867, 503, 134, 13, 1, 36, 406, 1792, 3332, 2914, 1279, 268, 21, 1, 45, 666, 4074, 10955, 13882, 9084, 3122, 527, 34, 1, 55, 1035, 8430, 31563

Offset: 1

Author

Clark Kimberling, Mar 29 2012

Keywords

• nonn
• tabl

Comments

Row n starts with 1, followed by the n-th triangular number, and ends with the n-th Fibonacci number.

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

Examples

			First five rows:
1
1...1
1...3....1
1...6....6....2
1...10...21...14...3
First three polynomials u(n,x): 1, 1 + x, 1 + 3x + x^2.
		

Crossrefs

Cf. A210867, 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] - 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[%]   (* A210866 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210867 *)

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)-x,

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