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

Offset

1,3

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.

Links

Table of n, a(n) for n=1..62.

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.

Example

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.
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+ 3x, 1 + 4x + 8x^2.

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 *)

Crossrefs

Cf. A210740, A208510.

Sequence in context: A217151 A081255 A005371 * A193605 A193667 A205878

Adjacent sequences: A210736 A210737 A210738 * A210740 A210741 A210742

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 24 2012

Status

approved