A210799 Triangle of coefficients of polynomials u(n,x) jointly generated with A210800; see the Formula section.

1, 3, 1, 5, 4, 2, 11, 13, 9, 3, 17, 32, 32, 17, 5, 35, 77, 96, 72, 32, 8, 53, 164, 254, 243, 153, 59, 13, 107, 353, 641, 739, 579, 313, 107, 21, 161, 704, 1496, 2042, 1938, 1305, 623, 192, 34, 323, 1433, 3440, 5348, 5898, 4774, 2831, 1213, 341, 55, 485

Offset: 1

Author

Clark Kimberling, Mar 27 2012

Keywords

• nonn
• tabl

Comments

Row n starts with A060647(n) and ends with F(n), where F=A000045 (Fibonacci numbers).

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

Examples

			First five rows:
1
3....1
5....4....2
11...13...9....3
17...32...32...17...5
First three polynomials u(n,x): 1, 3 + x, 5 + 4x + 2x^2.
		

Crossrefs

Cf. A210800, 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 = 1; 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[%]   (* A210799 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210800 *)

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) + a(k) with a(0) = 2, a(1) = -1, a(k) = 0 if k>1, T(1,0) = T(2,1) = 1, T(2,0) = 3 and T(n,k) = 0 if k<0 or if k>=n.