A210551 Triangle of coefficients of polynomials v(n,x) jointly generated with A172431; see the Formula section.

1, 3, 1, 5, 6, 1, 7, 15, 10, 1, 9, 28, 35, 15, 1, 11, 45, 84, 70, 21, 1, 13, 66, 165, 210, 126, 28, 1, 15, 91, 286, 495, 462, 210, 36, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 21, 190, 969, 3060, 6188

Offset: 1

Author

Clark Kimberling, Mar 22 2012

Keywords

• nonn
• tabl

Comments

Row sums: -1+odd-indexed Fibonacci numbers

Alternating row sums: 1,2,0,1,2,0,1,2,0,...

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

Examples

			From _Paul Weisenhorn_, May 17 2020 : (Start)
First five rows of v(n.x):
  1
  3   1
  5   6   1
  7  15  10   1
  9  28  35  15  1
First three polynomials v(n,x): 1, 3 + x, 5 + 6x + x^2. (End)
From _Paul Weisenhorn_, May 14 2020: (Start)
First five rows of u(n,x):
  1
  1  2
  1  4   3
  1  6  10   4
  1  8  21  20  5
First three polynomials u(n,x): 1, 1 + 2x, 1 + 4x + 3x^2. (End)
		

Crossrefs

Cf. A172431, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := u[n - 1, x] + (x + 1)*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[%]    (* A172431 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210551 *)

Formula

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

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

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