A210794 Triangle of coefficients of polynomials v(n,x) jointly generated with A210793; see the Formula section.

1, 1, 2, 3, 3, 3, 3, 11, 8, 5, 9, 18, 29, 17, 8, 9, 48, 67, 71, 35, 13, 27, 81, 180, 194, 158, 68, 21, 27, 189, 387, 575, 508, 338, 129, 34, 81, 324, 918, 1410, 1617, 1222, 695, 239, 55, 81, 702, 1890, 3606, 4471, 4222, 2793, 1393, 436, 89, 243, 1215

Offset: 1

Author

Clark Kimberling, Mar 26 2012

Keywords

• nonn
• tabl

Comments

Row n starts with a power of 3 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Row sums: A000244 (powers of 3)

Alternating row sums: A077925

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

Examples

			First five rows:
1
1...2
3...3....3
3...11...8....5
9...18...29...17...8
First three polynomials v(n,x): 1, 1 + 2x, 3 + 3x + 3x^2
		

Crossrefs

Cf. A210793, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 0; h = 2; p = -1; f = 0;
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[%]   (* A210793 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210794 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)

Formula

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

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

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

T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 29 2012