A210557 Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section.

1, 1, 2, 1, 3, 5, 1, 4, 10, 12, 1, 5, 16, 30, 29, 1, 6, 23, 56, 87, 70, 1, 7, 31, 91, 185, 245, 169, 1, 8, 40, 136, 334, 584, 676, 408, 1, 9, 50, 192, 546, 1158, 1784, 1836, 985, 1, 10, 61, 260, 834, 2052, 3850, 5312, 4925, 2378, 1, 11, 73, 341, 1212, 3366

Offset: 1

Author

Clark Kimberling, Mar 22 2012

Keywords

• nonn
• tabl

Comments

Row sums: powers of 3 (see A000244).

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

Subtriangle of (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 23 2012

Up to reflection at the vertical axis, this triangle coincides with the triangle given in A164981, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 20 2012

Examples

			First five rows:
  1;
  1, 2;
  1, 3,  5;
  1, 4, 10, 12;
  1, 5, 16, 30, 29;
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.
From _Philippe Deléham_, Mar 23 2012: (Start)
(1, 0, -1/2, 1/2, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, ...) begins:
  1;
  1, 0;
  1, 2,  0;
  1, 3,  5,  0;
  1, 4, 10, 12,  0;
  1, 5, 16, 30, 29, 0; (End)
		

Crossrefs

Cf. A210558, A208510, A164981.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%]   (* A210557 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210558 *)

Formula

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

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

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

From Philippe Deléham, Mar 23 2012. (Start)

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

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