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

Author

Clark Kimberling, Mar 24 2012

Keywords

• nonn
• tabf

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

Examples

			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)
		

Links

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

Crossrefs

Cf. A210596, A208510.

Programs

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 *)

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

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)