A209154 Triangle of coefficients of polynomials u(n,x) jointly generated with A209157; see the Formula section.

1, 2, 1, 4, 5, 2, 7, 14, 10, 2, 11, 32, 36, 18, 4, 16, 65, 104, 82, 32, 4, 22, 121, 258, 282, 168, 52, 8, 29, 210, 574, 820, 676, 328, 88, 8, 37, 344, 1176, 2116, 2264, 1504, 608, 136, 16, 46, 537, 2256, 4980, 6640, 5672, 3136, 1096, 224, 16, 56, 805

Offset: 1

Author

Clark Kimberling, Mar 07 2012

Keywords

• nonn
• tabl

Comments

Last number in each row is a power of 2.

Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,...

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

Examples

			First five rows:
1
2....1
4....5....2
7....14...10...2
11...32...36...18...4
First three polynomials v(n,x): 1, 2 + x, 4 + 5x + 2x^2.
		

Crossrefs

Cf. A209157, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 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[%]    (* A209154 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209157 *)

Formula

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

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

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