OFFSET
1,1
COMMENTS
Usually regarded as a square matrix T when combined with other matrices and column vectors.
Then T * V, where V = any sequence regarded as a column vector with offset 1 is a new sequence S [called an interpolation transform] given by S(2n) = V(2n), S(2n-1) = V(2n) + V(2n-1). Example: If T * [1,2,3,...], S = [1, 2, 5, 4, 9, 6, 13, 8, 17, ...) = A114752. A133080 is identical to A133566 except that the subdiagonal = (1,0,1,0,...). A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, 9, 19, ...).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,-1,0,0,0,0,0,0,...] DELTA [1,0,-2,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 15 2007
FORMULA
Odd rows: (n-2) zeros followed by 1, 1. Even rows: (n-1) zeros followed by 1.
Sum_{k=0..n} T(n,k) = A040001(n). - Philippe Deléham, Dec 15 2007
G.f.: (-1-x*y-x^2*y)*x*y/((-1+x*y)*(1+x*y)). - R. J. Mathar, Aug 11 2015
EXAMPLE
First few rows of the triangle:
1;
0, 1;
0, 1, 1;
0, 0, 0, 1;
0, 0, 0, 1, 1;
0, 0, 0, 0, 0, 1;
...
MAPLE
A133566 := proc(n, k)
if n = k then
1;
elif k=n-1 and type(n, odd) then
1;
else
0 ;
end if;
end proc: # R. J. Mathar, Jun 20 2015
MATHEMATICA
T[n_, k_] := Which[n == k, 1, k == n - 1 && OddQ[n], 1, True, 0];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 24 2023 *)
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Sep 16 2007
EXTENSIONS
Entry revised by N. J. A. Sloane, Jun 20 2015
STATUS
approved