A210221 Triangle of coefficients of polynomials u(n,x) jointly generated with A210596; see the Formula section.

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

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