A207613 Triangle of coefficients of polynomials v(n,x) jointly generated with A207612; see Formula section.

1, 2, 2, 3, 4, 4, 5, 8, 8, 8, 8, 16, 20, 16, 16, 13, 30, 44, 48, 32, 32, 21, 56, 92, 112, 112, 64, 64, 34, 102, 188, 256, 272, 256, 128, 128, 55, 184, 372, 560, 672, 640, 576, 256, 256, 89, 328, 724, 1184, 1552, 1696, 1472, 1280, 512, 512, 144, 580, 1384

Offset: 1

Author

Clark Kimberling, Feb 19 2012

Keywords

• nonn
• tabl
• easy

Comments

Only column 1 contains odd numbers.

column 1: A000045 (Fibonacci sequence)

row sums: A002878 (bisection of Lucas sequence)

top edge: A000079 (powers of 2)

Examples

			First five rows:
  1
  2  2
  3  4  4
  5  8  8  8
  8 16 20 16 16
		

Crossrefs

A000045 (column 1), A000079 (main diagonal), A002878 (row sums). Cf. A207612, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207612 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207613 *)

Formula

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

With offset 0, the Riordan array ((1 + z)/(1 - z - z^2), 2*z*(1 - z)/(1 - z - z^2)) with o.g.f. (1 + z)/(1 - z - z^2 - x*(2*z - 2*z^2)) = 1 + (2 + 2*x)*z + (3 + 4*x + 4*x^2)*z^2 + .... - Peter Bala, Dec 30 2015