A208508 Triangle of coefficients of polynomials u(n,x) jointly generated with A208509; see the Formula section.

1, 1, 1, 1, 4, 1, 9, 1, 1, 16, 6, 1, 25, 20, 1, 1, 36, 50, 8, 1, 49, 105, 35, 1, 1, 64, 196, 112, 10, 1, 81, 336, 294, 54, 1, 1, 100, 540, 672, 210, 12, 1, 121, 825, 1386, 660, 77, 1, 1, 144, 1210, 2640, 1782, 352, 14, 1, 169, 1716, 4719, 4290, 1287, 104, 1, 1

Offset: 1

Author

Clark Kimberling, Feb 27 2012

Keywords

• nonn
• tabf

Comments

col 1: A000012

col 2: A000290 (squares)

col 3: A002415

col 4: A040977

col 5: A054334

row sums, u(n,1): A083329

Examples

			First five rows:
1
1...1
1...4
1...9....1
1...16...6
First five polynomials u(n,x):
1
1 + x
1 + 4x
1 + 9x + x^2
1 + 16x + 6x^2
		

Crossrefs

Cf. A208509.

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] + 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[%]    (* A208508 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208509 *)

Formula

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

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

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