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

1,6

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

Links

Table of n, a(n) for n=1..69.

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)

Example

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)

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 *)

Crossrefs

Cf. A208329.

Sequence in context: A211315 A096583 A130154 * A134398 A026615 A026681

Adjacent sequences: A208325 A208326 A208327 * A208329 A208330 A208331

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved