A208522 Triangle of coefficients of polynomials u(n,x) jointly generated with A208522; see the Formula section.

1, 1, 1, 1, 2, 3, 1, 3, 6, 5, 1, 4, 9, 12, 11, 1, 5, 12, 21, 30, 21, 1, 6, 15, 32, 57, 64, 43, 1, 7, 18, 45, 92, 133, 146, 85, 1, 8, 21, 60, 135, 232, 329, 316, 171, 1, 9, 24, 77, 186, 365, 612, 765, 694, 341, 1, 10, 27, 96, 245, 536, 1015, 1512, 1801, 1496, 683

Offset: 1

Author

Clark Kimberling, Feb 29 2012

Keywords

• nonn
• tabl

Examples

			First five rows:
1
1...1
1...2...3
1...3...6...5
1...4...9...12...11
First five polynomials u(n,x):
1
1 + x
1 + 2x + 3x^2
1 + 3x + 6x^2 + 5x^3
1 + 4x + 9x^2 + 12x^3 + 11x^4
		

Crossrefs

Cf. A208523.

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_] := 2 x*u[n - 1, x] + 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[%]    (* A208522 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208523 *)

Formula

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

v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,

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