OFFSET
1,3
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
LINKS
G. C. Greubel, Rows n=1..102 of triangle, flattened
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
EXAMPLE
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.
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 *)
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(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print (a(n)) # Indranil Ghosh, May 27, 2017
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 24 2012
STATUS
approved