A209145 Triangle of coefficients of polynomials u(n,x) jointly generated with A122075; see the Formula section.

1, 2, 1, 4, 4, 1, 7, 10, 5, 1, 12, 22, 16, 6, 1, 20, 45, 43, 23, 7, 1, 33, 88, 104, 72, 31, 8, 1, 54, 167, 235, 199, 110, 40, 9, 1, 88, 310, 506, 506, 340, 158, 50, 10, 1, 143, 566, 1051, 1211, 956, 538, 217, 61, 11, 1, 232, 1020, 2123, 2768, 2507, 1652, 805, 288, 73, 12, 1

Offset: 1

Author

Clark Kimberling, Mar 06 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
  1
  2    1
  4    4    1
  7    10   5    1
  12   22   16   6   1
First three polynomials u(n,x): 1, 2 + x, 4 + 4*x + x^2.
		

Crossrefs

Cf. A209144, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 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[%]    (* A209145 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A122075 *)

Formula

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

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

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