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A133566
Triangle read by rows: (1,1,1,...) on the main diagonal and (0,1,0,1,...) on the subdiagonal.
14
1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
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
nonn,tabl,easy
AUTHOR
Gary W. Adamson, Sep 16 2007
EXTENSIONS
Entry revised by N. J. A. Sloane, Jun 20 2015
STATUS
approved