OFFSET
1,2
COMMENTS
As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012
FORMULA
u(n,x) = u(n-1,x) + v(n-1,x),
v(n,x) = 2x*u(n-1,x) + (x+1)v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 03 2012: (Start)
As triangle T(n,k), 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-1,k) + T(n-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-y*x)/(1 - (2+y)*x - (y-1)*x^2).
EXAMPLE
First five rows:
1;
2;
3, 3;
4, 11, 3;
5, 26, 20, 3;
Triangle (2, -1/2, 1/2, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, ...) begins:
1;
2, 0;
3, 3, 0;
4, 11, 3, 0;
5, 26, 20, 3, 0;
6, 50, 74, 29, 3, 0;
7, 85, 204, 149, 38, 3, 0;
... - Philippe Deléham, Mar 03 2012
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_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A207608 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A207609 *)
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 2*x*u(n - 1, x) + (x + 1)*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 28 2017
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Feb 19 2012
STATUS
approved