A208328 Triangle of coefficients of polynomials u(n,x) jointly generated with A208329; see the Formula section.

1, 1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 9, 11, 1, 1, 9, 13, 25, 21, 1, 1, 11, 17, 43, 53, 43, 1, 1, 13, 21, 65, 97, 125, 85, 1, 1, 15, 25, 91, 153, 255, 273, 171, 1, 1, 17, 29, 121, 221, 441, 597, 609, 341, 1, 1, 19, 33, 155, 301, 691, 1089, 1443, 1325, 683, 1, 1, 21

Offset: 1

Author

Clark Kimberling, Feb 26 2012

Keywords

• nonn
• tabl

Comments

Row sums, u(n,1): A000129

Row sums, v(n,1): A001333

Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 07 2012

Examples

			First five rows:
  1;
  1, 1;
  1, 1, 3;
  1, 1, 5, 5;
  1, 1, 7, 9, 11;
First five polynomials u(n,x):
  1
  1 + x
  1 + x + 3x^2
  1 + x + 5x^2 + 5x^3
  1 + x + 7x^2 + 9x^3 + 11x^4.
From _Philippe Deléham_, Mar 07 2012: (Start)
(1, 0, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, ...) begins:
  1;
  1, 0;
  1, 1,  0;
  1, 1,  3,  0;
  1, 1,  5,  5,  0;
  1, 1,  7,  9, 11,  0;
  1, 1,  9, 13, 25, 21,  0;
  1, 1, 11, 17, 43, 53, 43, 0; (End)
		

Crossrefs

Cf. A208329.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*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[%]  (* A208328 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208329 *)

Formula

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

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

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

From Philippe Deléham, Mar 07 2012: (Start)

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

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

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

Sum_{k=0..n, n>0} T(n,k)*x^k = A000012(n), A000129(n), A083858(n) for x = 0, 1, 2 respectively. (End)