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

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