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

1,4

Comments

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

Links

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A208510 A131767 A026780 * A320435 A275421 A243576

Adjacent sequences: A209418 A209419 A209420 * A209422 A209423 A209424

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Mar 09 2012

Status

approved