A208765 Triangle of coefficients of polynomials u(n,x) jointly generated with A208766; see the Formula section.

1, 1, 2, 1, 4, 6, 1, 6, 18, 14, 1, 8, 36, 56, 38, 1, 10, 60, 140, 190, 94, 1, 12, 90, 280, 570, 564, 246, 1, 14, 126, 490, 1330, 1974, 1722, 622, 1, 16, 168, 784, 2660, 5264, 6888, 4976, 1606, 1, 18, 216, 1176, 4788, 11844, 20664, 22392, 14454, 4094, 1

Offset: 1

Author

Clark Kimberling, Mar 02 2012

Keywords

• nonn
• tabl

Comments

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

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

Examples

			First five rows:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 18, 14;
  1,  8, 36, 56, 38;
First five polynomials u(n,x):
  1
  1 + 2x
  1 + 4x + 6x^2
  1 + 6x + 18x^2 + 14x^3
  1 + 8x + 36x^2 + 56x^3 + 38x^4
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 2, 1, -2, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  4,  6,   0;
  1,  6, 18,  14,   0;
  1,  8, 36,  56,  38,  0;
  1, 10, 60, 140, 190, 94, 0. - _Philippe Deléham_, Mar 18 2012
		

Links

• G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Crossrefs

Cf. A208766, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 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[%]     (* A208765 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A208766 *)
Rest[CoefficientList[CoefficientList[Series[(1-x-y*x+2*y*x^2-4*y^2*x^2)/( 1-2*x-y*x+x^2+y*x^2-4*y^2*x^2), {x,0,20}, {y,0,20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)

Formula

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

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

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

From Philippe Deléham, Mar 18 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

G.f.: (1-x-y*x+2*y*x^2-4*y^2*x^2)/(1-2*x-y*x+x^2+y*x^2-4*y^2*x^2).

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 4*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.

T(n,k) = binomial(n-1,k)*A026597(k). (End)