A208511 Triangle of coefficients of polynomials u(n,x) jointly generated with A208512; see the Formula section.

1, 1, 1, 1, 3, 2, 1, 5, 7, 4, 1, 7, 14, 16, 8, 1, 9, 23, 37, 36, 16, 1, 11, 34, 69, 94, 80, 32, 1, 13, 47, 114, 195, 232, 176, 64, 1, 15, 62, 174, 354, 528, 560, 384, 128, 1, 17, 79, 251, 588, 1041, 1384, 1328, 832, 256, 1, 19, 98, 347, 916, 1863, 2938, 3536

Offset: 1

Author

Clark Kimberling, Feb 28 2012

Keywords

• nonn
• tabl

Examples

			First five rows:
1
1...1
1...3...2
1...5...7....4
1...7...14...16...8
First five polynomials u(n,x):
1
1 + x
1 + 3x + 2x^2
1 + 5x + 7x^2 + 4x^3
1 + 7x + 14x^2 + 16x^3 + 8x^4
		

Crossrefs

Cf. A208511.

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_] := u[n - 1, x] + 2 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[%]  (* A208511 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208512 *)

Formula

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

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

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