A208606 Triangle of coefficients of polynomials u(n,x) jointly generated with A208607; see the Formula section.

1, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 16, 12, 7, 1, 1, 25, 30, 26, 7, 1, 1, 36, 65, 73, 32, 10, 1, 1, 49, 126, 175, 113, 52, 10, 1, 1, 64, 224, 378, 332, 199, 61, 13, 1, 1, 81, 372, 756, 852, 634, 277, 87, 13, 1, 1, 100, 585, 1422, 1974, 1779, 1024, 421, 99, 16, 1, 1, 121

Offset: 1

Author

Clark Kimberling, Feb 29 2012

Keywords

• nonn
• tabl

Examples

			First five rows:
1
1...1
1...4....1
1...9....4....1
1...16...12...7...1
First five polynomials u(n,x):
1
1 + x
1 + 4x + 1x^2
1 + 9x + 4x^2 + x^3
1 + 16x + 12x^2 + 7x^3 + x^4
		

Crossrefs

Cf. A208607.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*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[%]  (* A208606 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208607 *)

Formula

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

Extensions

a(68) corrected by Georg Fischer, Sep 03 2021