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A209422 Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section. 3
1, 3, 5, 1, 9, 2, 1, 15, 6, 2, 1, 25, 13, 7, 2, 1, 41, 28, 16, 8, 2, 1, 67, 56, 38, 19, 9, 2, 1, 109, 109, 82, 49, 22, 10, 2, 1, 177, 206, 173, 112, 61, 25, 11, 2, 1, 287, 382, 352, 252, 146, 74, 28, 12, 2, 1, 465, 697, 701, 543, 347, 184, 88, 31, 13, 2, 1, 753, 1256, 1368, 1144, 784, 459, 226, 103, 34, 14, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

FORMULA

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

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

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

G.f.: (1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)) = 1 + 3*t + (5 + x)*t^2 + ... . - G. C. Greubel, Jan 03 2018

EXAMPLE

First five rows:

   1;

   3;

   5, 1;

   9, 2, 1;

  15, 6, 2, 1;

First three polynomials v(n,x): 1, 3, 5 + x.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];

v[n_, x_] := u[n - 1, 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[%]    (* A209421 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A209422 *)

CoefficientList[CoefficientList[Series[(1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)), {t, 0, 10}], t], x]// Flatten (* G. C. Greubel, Jan 03 2018 *)

CROSSREFS

Cf. A209421, A208510.

Sequence in context: A197326 A235605 A212695 * A320386 A112411 A283838

Adjacent sequences:  A209419 A209420 A209421 * A209423 A209424 A209425

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Mar 09 2012

STATUS

approved

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Last modified October 16 08:15 EDT 2019. Contains 328051 sequences. (Running on oeis4.)