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A209421 Triangle of coefficients of polynomials u(n,x) jointly generated with A209422; see the Formula section. 3
1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 7, 5, 1, 1, 15, 15, 9, 6, 1, 1, 25, 28, 22, 11, 7, 1, 1, 41, 53, 44, 30, 13, 8, 1, 1, 67, 97, 91, 63, 39, 15, 9, 1, 1, 109, 176, 179, 140, 85, 49, 17, 10, 1, 1, 177, 315, 349, 291, 201, 110, 60, 19, 11, 1, 1, 287, 559, 667, 601, 437, 275, 138, 72, 21, 12, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

Riordan array (f(z), z*g(z)) where f(z) = (1 - z + z^2)/(1 - 2*z + z^3) is the o.g.f. for A001595 and g(z) = (1 - z)/(1 - z - z^2) is the o.g.f. for A212804, a variant of the Fibonacci numbers. - Peter Bala, Dec 30 2015

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

EXAMPLE

First five rows:

  1

  1 1

  3 1 1

  5 4 1 1

  9 7 5 1 1

First three polynomials v(n,x): 1, 1 + x, 3 + x + x^2.

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 - 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. A001595 (column 1), A209422, A208510, A212804.

Sequence in context: A208510 A131767 A026780 * A320435 A275421 A243576

Adjacent sequences:  A209418 A209419 A209420 * A209422 A209423 A209424

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Mar 09 2012

STATUS

approved

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Last modified October 19 03:08 EDT 2019. Contains 328211 sequences. (Running on oeis4.)