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

1, 1, 2, 2, 2, 4, 3, 6, 4, 8, 5, 10, 16, 8, 16, 8, 20, 28, 40, 16, 32, 13, 36, 64, 72, 96, 32, 64, 21, 66, 124, 184, 176, 224, 64, 128, 34, 118, 248, 376, 496, 416, 512, 128, 256, 55, 210, 476, 808, 1056, 1280, 960, 1152, 256, 512, 89, 370, 908, 1640, 2416

Offset: 1

Author

Clark Kimberling, Mar 24 2012

Keywords

• nonn
• tabl

Comments

Row n begins with F(n) and ends with 2^(n-1), where F = A000045 (Fibonacci numbers)

Row sums: odd-indexed Fibonacci numbers, see A001519.

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

Riordan array (1/(1 - z - z^2), 2*z*(1 - z)/(1 - z - z^2)). - Peter Bala, Dec 30 2015

Examples

			First five rows:
  1
  1  2
  2  2  4
  3  6  4  8
  5 10 16  8 16
First three polynomials v(n,x): 1, 1 + 2x, 2 + 2x + 4x^2.
		

Links

• G. C. Greubel, Rows n=1..102 of triangle, flattened

Crossrefs

Cf. A000045, A000079, A001519, A210221, 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}, CoefficientList[CoefficientList[Series[1/((1-x-x^2) - t*2*x*(1-x)), {x, 0, m}, {t, 0, m}], x], t]]//Flatten (* G. C. Greubel, Dec 15 2018 *)

PARI

{T(n,k) = if(n==1 && k==0, 1, if(n==2 && k==0, 1, if(n==2 && k==1, 2, if(k<0 || k>n-1, 0, T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1) ))))};
for(n=1,15, for(k=0, n-1, print1(T(n,k), ", "))) \\ G. C. Greubel, Dec 15 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(v(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) + 2x*v(n-1,x),

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 25 2012

G.f.: 1/((1-x-x^2) - t*2*x*(1-x)). - G. C. Greubel, Dec 15 2018