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%I #18 Sep 08 2022 08:45:41
%S 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,
%T 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,
%U 0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,2,1
%N Triangle by columns: (1, 2, 0, 0, 0, ...) in every column.
%C Binomial transform of the triangle = A110813.
%C Eigensequence of the triangle = A001045
%C Inverse = a triangle with (1, -2, 4, -8, 16, ...) in every column.
%C 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
%H G. C. Greubel, <a href="/A156319/b156319.txt">Rows n = 1..100 of triangle, flattened</a>
%F Triangle read by rows, T(n,k) = 1 if n=k, 2 if k = n-1, 0 otherwise.
%F By columns, (1, 2, 0, 0, 0, ...) in every column.
%F T(n,k) = A097806(n,k)*2^(n-k). - _Philippe Deléham_, Feb 08 2009
%F G.f.: (1+2*x)*x*y/(1-x*y). - _R. J. Mathar_, Aug 12 2015
%e First few rows of the triangle:
%e 1;
%e 2, 1;
%e 0, 2, 1;
%e 0, 0, 2, 1;
%e 0, 0, 0, 2, 1;
%e 0, 0, 0, 0, 2, 1;
%e 0, 0, 0, 0, 0, 2, 1;
%e 0, 0, 0, 0, 0, 0, 2, 1;
%e 0, 0, 0, 0, 0, 0, 0, 2, 1;
%e ...
%p T:= proc (n) option remember;
%p if k=n then 1
%p elif k=n-1 then 2
%p else 0 fi;
%p end proc;
%p seq(seq(T(n,k), k=1..n), n = 1..15); # _G. C. Greubel_, Sep 20 2019
%t Table[If[k==n,1, If[k==n-1, 2, 0]], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Sep 20 2019 *)
%t Join[{1},Flatten[Table[PadRight[{2,1},n,0],{n,3,20}]]] (* _Harvey P. Dale_, Feb 28 2022 *)
%o (PARI) T(n,k) = if(k==n, 1, if(k==n-1, 2, 0)); \\ _G. C. Greubel_, Sep 20 2019
%o (Magma) T:= func< n,k | k eq n select 1 else k eq n-1 select 2 else 0 >;
%o [T(n,k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Sep 20 2019
%o (Sage)
%o def T(n,k):
%o if (k==n): return 1
%o elif (k==n-1): return 2
%o else: return 0
%o [[T(n,k) for k in (1..n)] for n in (1..15)] # _G. C. Greubel_, Sep 20 2019
%o (GAP)
%o T:= function(n,k)
%o if k=n then return 1;
%o elif k=n-1 then return 2;
%o else return 0;
%o fi;
%o end;
%o Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # _G. C. Greubel_, Sep 20 2019
%Y Cf. A001045, A097806, A110813.
%K nonn,tabl,easy
%O 1,2
%A _Gary W. Adamson_, Feb 07 2009
%E More terms added by _G. C. Greubel_, Sep 20 2019
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