A156319 Triangle by columns: (1, 2, 0, 0, 0, ...) in every column.

1, 2, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1

Offset

1,2

Comments

Binomial transform of the triangle = A110813.

Eigensequence of the triangle = A001045

Inverse = a triangle with (1, -2, 4, -8, 16, ...) in every column.

Triangle T(n,k), 0 <= k <= n, given by [2,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 08 2009

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

Triangle read by rows, T(n,k) = 1 if n=k, 2 if k = n-1, 0 otherwise.

By columns, (1, 2, 0, 0, 0, ...) in every column.

T(n,k) = A097806(n,k)*2^(n-k). - Philippe Deléham, Feb 08 2009

G.f.: (1+2*x)*x*y/(1-x*y). - R. J. Mathar, Aug 12 2015

Example

First few rows of the triangle:
  1;
  2, 1;
  0, 2, 1;
  0, 0, 2, 1;
  0, 0, 0, 2, 1;
  0, 0, 0, 0, 2, 1;
  0, 0, 0, 0, 0, 2, 1;
  0, 0, 0, 0, 0, 0, 2, 1;
  0, 0, 0, 0, 0, 0, 0, 2, 1;
...

Maple

T:= proc (n) option remember;
if k=n then 1
elif k=n-1 then 2
else 0 fi;
end proc;
seq(seq(T(n, k), k=1..n), n = 1..15); # G. C. Greubel, Sep 20 2019

Mathematica

Table[If[k==n, 1, If[k==n-1, 2, 0]], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Sep 20 2019 *)
Join[{1}, Flatten[Table[PadRight[{2, 1}, n, 0], {n, 3, 20}]]] (* Harvey P. Dale, Feb 28 2022 *)

Prog

(PARI) T(n, k) = if(k==n, 1, if(k==n-1, 2, 0)); \\ G. C. Greubel, Sep 20 2019
(Magma) T:= func< n, k | k eq n select 1 else k eq n-1 select 2 else 0 >;
[T(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Sep 20 2019
(Sage)
def T(n, k):
    if (k==n): return 1
    elif (k==n-1): return 2
    else: return 0
[[T(n, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Sep 20 2019
(GAP)
T:= function(n, k)
    if k=n then return 1;
    elif k=n-1 then return 2;
    else return 0;
    fi;
  end;
Flat(List([1..15], n-> List([1..n], k-> T(n, k) ))); # G. C. Greubel, Sep 20 2019

Crossrefs

Cf. A001045, A097806, A110813.

Sequence in context: A058531 A093073 A251635 * A190893 A030204 A083650

Adjacent sequences: A156316 A156317 A156318 * A156320 A156321 A156322

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Feb 07 2009

Extensions

More terms added by G. C. Greubel, Sep 20 2019

Status

approved