A209414 Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.

1, 1, 1, 1, 4, 1, 1, 7, 9, 1, 1, 10, 26, 16, 1, 1, 13, 52, 70, 25, 1, 1, 16, 87, 190, 155, 36, 1, 1, 19, 131, 403, 553, 301, 49, 1, 1, 22, 184, 736, 1462, 1372, 532, 64, 1, 1, 25, 246, 1216, 3206, 4446, 3024, 876, 81, 1, 1, 28, 317, 1870, 6190, 11584, 11826, 6084, 1365, 100, 1

Offset

1,5

Comments

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

Subtriangle of the triangle given by (1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012

Links

G. C. Greubel, Table of n, a(n) for n = 1..1225

Formula

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

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

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

From Philippe Deléham, Apr 01 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1-2*y*x-2*y*x^2+y^2*x^2)/(1-x-2*y*x-y*x^2+y^2*x^2).

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)

Example

First five rows:
  1;
  1,  1;
  1,  4,  1;
  1,  7,  9,  1;
  1, 10, 26, 16,  1;
First three polynomials v(n,x):
  1
  1 + x
  1 + 4x + x^2.
From Philippe Deléham, Apr 01 2012: (Start)
(1, 0, 2, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  1,  0;
  1,  4,  1,  0;
  1,  7,  9,  1,  0;
  1, 10, 26, 16,  1,  0;
  1, 13, 52, 70, 25,  1,  0; (End)

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_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
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[%]    (* A209414 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A112351 *)
CoefficientList[CoefficientList[Series[(1 - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - x - 2*y*x - y*x^2 + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

Crossrefs

Cf. A112351, A208510.

Sequence in context: A316123 A146771 A073697 * A193636 A232968 A119673

Adjacent sequences: A209411 A209412 A209413 * A209415 A209416 A209417

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 09 2012

Status

approved