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A207613 Triangle of coefficients of polynomials v(n,x) jointly generated with A207612; see Formula section. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)