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A210221 Triangle of coefficients of polynomials u(n,x) jointly generated with A210596; see the Formula section. 4
1, 2, 3, 2, 5, 4, 4, 8, 10, 8, 8, 13, 20, 24, 16, 16, 21, 40, 52, 56, 32, 32, 34, 76, 116, 128, 128, 64, 64, 55, 142, 240, 312, 304, 288, 128, 128, 89, 260, 488, 688, 800, 704, 640, 256, 256, 144, 470, 964, 1496, 1856, 1984, 1600, 1408, 512, 512, 233, 840 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums: even-indexed Fibonacci numbers.

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

Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

LINKS

G. C. Greubel, Rows n = 1..100 of triangle, flattened

FORMULA

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

v(n,x) = u(n-1,x) + 2*x*v(n-1,x) [Corrected by Indranil Ghosh, May 27 2017]

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

From Philippe Deléham, Mar 25 2012: (Start)

As DELTA-triangle T(n,k) with 0 <= k <= n:

G.f.: (1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2).

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 2, T(1,1) = T(2,1) = T(2,2) = 0, T(n,k) = 0 if k < 0 or if k >= n. (End)

EXAMPLE

First five rows:

  1;

  2;

  3,  2;

  5,  4, 4;

  8, 10, 8, 8;

First three polynomials u(n,x):

  1

  2

  3 + 2x.

From Philippe Deléham, Mar 25 2012: (Start)

(1, 1, -1, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, ...) begins:

   1;

   1,  0;

   2,  0,  0;

   3,  2,  0,  0;

   5,  4,  4,  0,  0;

   8, 10,  8,  8,  0,  0;

  13, 20, 24, 16, 16,  0,  0; (End)

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];

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[%]   (* A210221 *)

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

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

TableForm[cv]

Flatten[%]   (* A210596 *)

With[{m = 10}, Rest[CoefficientList[CoefficientList[Series[(1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2), {x, 0, m}, {y, 0, m}], x], y]]]//Flatten (* G. C. Greubel, Dec 16 2018 *)

T[n_, k_]:= If[k < 0 || k > n, 0, T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k] - 2*T[n-2, k-1]]; T[1, 0] = 1 ; T[2, 0] = 2; T[2, 1] = 0; Join[{1}, Table[T[n, k], {n, 1, 10}, {k, 0, n-2}]//Flatten] (* G. C. Greubel, Dec 17 2018 *)

PROG

(Python)

from sympy import Poly

from sympy.abc import x

def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)

def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x)

def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]

for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

CROSSREFS

Cf. A210596, A208510.

Sequence in context: A089587 A278327 A067316 * A232113 A216475 A267807

Adjacent sequences:  A210218 A210219 A210220 * A210222 A210223 A210224

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Mar 24 2012

STATUS

approved

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Last modified October 5 23:01 EDT 2022. Contains 357261 sequences. (Running on oeis4.)