A210739 Triangle of coefficients of polynomials u(n,x) jointly generated with A210740; see the Formula section.

1, 1, 3, 1, 4, 8, 1, 4, 14, 21, 1, 4, 15, 46, 55, 1, 4, 15, 55, 145, 144, 1, 4, 15, 56, 196, 444, 377, 1, 4, 15, 56, 208, 678, 1331, 987, 1, 4, 15, 56, 209, 764, 2282, 3926, 2584, 1, 4, 15, 56, 209, 779, 2762, 7499, 11434, 6765, 1, 4, 15, 56, 209, 780, 2892, 9804, 24141, 32960, 17711

Offset: 1

Author

Clark Kimberling, Mar 24 2012

Keywords

• nonn
• tabl

Comments

Rows end with even-indexed Fibonacci numbers.

Limiting row: A001353

Row sums: A003562

Alternating row sums: A000975 (signed)

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

Examples

			First five rows:
  1
  1   3
  1   4   8
  1   4   14   21
  1   4   15   46   55
First three polynomials u(n,x): 1, 1 + 3*x, 1 + 4*x + 8*x^2.
		

Crossrefs

Cf. A210740, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]    (* A210739 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210740 *)

Formula

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

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

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