OFFSET
1,2
COMMENTS
Column 1: Fibonacci numbers, A000045
Column 2: A029907
Row sums: A003945.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012
FORMULA
u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + x*v(n-1,x)+1, where u(1,x)=1, v(1,x)=1.
T(n,k) = T(n-1,k) + (n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k >= n.
EXAMPLE
First five rows:
1;
2, 1;
3, 2, 1;
5, 4, 2, 1;
8, 8, 5, 2, 1;
From Philippe Deléham, Mar 25 2012: (Start)
(0, 2, -1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, ...) begins:
1;
0, 1;
0, 2, 1;
0, 3, 2, 1;
0, 5, 4, 2, 1;
0, 8, 8, 5, 2, 1;
0, 13, 15, 11, 6, 2, 1;
0, 21, 28, 23, 14, 7, 2, 1; (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] + x*v[n - 1, x] + 1
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[%] (* A207610 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A207611 *)
T[ n_, k_] := Which[k<0 || n<0, 0, n<2, Boole[k<=n] + Boole[k==0&&n==1], True, T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* Michael Somos, Sep 19 2024 *)
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) + x*v(n - 1, x) + 1
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017
(PARI) {T(n, k) = if(k<0 || n<0, 0, n<2, (k<=n) + (k==0 && n==1), T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* Michael Somos, Sep 19 2024 */
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 19 2012
STATUS
approved