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

Author

Clark Kimberling, Mar 09 2012

Keywords

• nonn
• tabl

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

Examples

			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)
		

Links

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

Crossrefs

Cf. A112351, 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_] := 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 *)

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)