A209422 Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section.

1, 3, 5, 1, 9, 2, 1, 15, 6, 2, 1, 25, 13, 7, 2, 1, 41, 28, 16, 8, 2, 1, 67, 56, 38, 19, 9, 2, 1, 109, 109, 82, 49, 22, 10, 2, 1, 177, 206, 173, 112, 61, 25, 11, 2, 1, 287, 382, 352, 252, 146, 74, 28, 12, 2, 1, 465, 697, 701, 543, 347, 184, 88, 31, 13, 2, 1, 753, 1256, 1368, 1144, 784, 459, 226, 103, 34, 14, 2

Offset: 1

Author

Clark Kimberling, Mar 09 2012

Keywords

• nonn
• tabf

Comments

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

Examples

			First five rows:
   1;
   3;
   5, 1;
   9, 2, 1;
  15, 6, 2, 1;
First three polynomials v(n,x): 1, 3, 5 + x.
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A209421, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, 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[%]    (* A209421 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209422 *)
CoefficientList[CoefficientList[Series[(1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)), {t, 0, 10}], t], x]// Flatten (* G. C. Greubel, Jan 03 2018 *)

Formula

u(n,x) = x*u(n-1,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.

G.f.: (1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)) = 1 + 3*t + (5 + x)*t^2 + ... . - G. C. Greubel, Jan 03 2018