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

1,2

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)

Links

Table of n, a(n) for n=1..58.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A097793 A015742 A015754 * A321424 A289677 A113967

Adjacent sequences: A207610 A207611 A207612 * A207614 A207615 A207616

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Feb 19 2012

Status

approved