A209421 Triangle of coefficients of polynomials u(n,x) jointly generated with A209422; see the Formula section.

1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 7, 5, 1, 1, 15, 15, 9, 6, 1, 1, 25, 28, 22, 11, 7, 1, 1, 41, 53, 44, 30, 13, 8, 1, 1, 67, 97, 91, 63, 39, 15, 9, 1, 1, 109, 176, 179, 140, 85, 49, 17, 10, 1, 1, 177, 315, 349, 291, 201, 110, 60, 19, 11, 1, 1, 287, 559, 667, 601, 437, 275, 138, 72, 21, 12, 1, 1

Offset: 1

Author

Clark Kimberling, Mar 09 2012

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			First five rows:
  1
  1 1
  3 1 1
  5 4 1 1
  9 7 5 1 1
First three polynomials v(n,x): 1, 1 + x, 3 + x + x^2.
		

Links

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

Crossrefs

Cf. A001595 (column 1), A209422, A208510, A212804.

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 - 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.

Riordan array (f(z), z*g(z)) where f(z) = (1 - z + z^2)/(1 - 2*z + z^3) is the o.g.f. for A001595 and g(z) = (1 - z)/(1 - z - z^2) is the o.g.f. for A212804, a variant of the Fibonacci numbers. - Peter Bala, Dec 30 2015

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