A210196 Triangle of coefficients of polynomials v(n,x) jointly generated with A210195; see the Formula section.

1, 1, 4, 1, 8, 8, 1, 12, 24, 16, 1, 16, 48, 64, 32, 1, 20, 80, 160, 160, 64, 1, 24, 120, 320, 480, 384, 128, 1, 28, 168, 560, 1120, 1344, 896, 256, 1, 32, 224, 896, 2240, 3584, 3584, 2048, 512, 1, 36, 288, 1344, 4032, 8064, 10752, 9216, 4608, 1024, 1, 40

Offset: 1

Author

Clark Kimberling, Mar 18 2012

Keywords

• nonn
• tabl

Comments

Periodic alternating row sums: 1, -3, 1, -3, 1, -3, ...

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

Examples

			First five rows:
  1;
  1,  4;
  1,  8,  8;
  1, 12, 24, 16;
  1, 16, 48, 64, 32;
First three polynomials v(n,x): 1, 1 + 4x, 1 + 8x + 8x^2.
		

Crossrefs

Cf. A134347, A210195, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := 2 x*u[n - 1, x] + 2 x*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[%]   (* A210195 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210196 *)

Formula

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

Conjecture: T(n,0) = 1 and T(n,k) = 2^(k+1)*binomial(n-1,k) if k>0. - Knud Werner, Jan 10 2022