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

Author

Gary W. Adamson, Feb 07 2009

Keywords

• nonn
• tabl
• easy

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

Examples

			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;
...
		

Links

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

Crossrefs

Cf. A001045, A097806, A110813.

Programs

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

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

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 *)

PARI

T(n,k) = if(k==n, 1, if(k==n-1, 2, 0)); \\ 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

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

Extensions

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