A208761 Triangle of coefficients of polynomials u(n,x) jointly generated with A208762; see the Formula section.

1, 1, 2, 1, 6, 4, 1, 12, 18, 8, 1, 20, 52, 50, 16, 1, 30, 120, 186, 126, 32, 1, 42, 240, 534, 576, 306, 64, 1, 56, 434, 1302, 1986, 1654, 718, 128, 1, 72, 728, 2828, 5712, 6632, 4484, 1650, 256, 1, 90, 1152, 5628, 14436, 21912, 20508, 11682, 3726, 512

Offset: 1

Author

Clark Kimberling, Mar 03 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,...

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

Subtriangle of the triangle given by [1,0,1,0,0,0,0,...] DELTA [0,2,0,-1,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

Examples

			First five rows:
  1;
  1,  2;
  1,  6,  4;
  1, 12, 18,  8;
  1, 20, 52, 50, 16;
First five polynomials u(n,x):
  1
  1 +  2x
  1 +  6x +  4x^2
  1 + 12x + 18x^2 +  8x^3
  1 + 20x + 52x^2 + 50x^3 + 16x^4
From _Philippe Deléham_, Mar 04 2012: (Start)
Triangle (1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, ...) begins:
  1;
  1,   0;
  1,   2,   0;
  1,   6,   4,   0;
  1,  12,  18,   8,   0;
  1,  20,  52,  50,  16,   0;
  1,  30, 120, 186, 126,  32,  0; (End)
		

Crossrefs

Cf. A208762, 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_] := (x + 1)*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[%]    (* A208761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208762 *)

Formula

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

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

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

Recurrence: T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 04 2012

G.f.: (-1-x*y+x)*x*y/(-1+x*y+2*x+2*x^2*y^2+x^2*y-x^2). - R. J. Mathar, Aug 12 2015