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A128174 Transform, (1,0,1,...) in every column.

%I A128174 #36 Sep 08 2022 08:45:30
%S A128174 1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,
%T A128174 0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,
%U A128174 0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1
%N A128174 Transform, (1,0,1,...) in every column.
%C A128174 Inverse of the triangle = a tridiagonal matrix with (1,1,1,...) in the superdiagonal, (0,0,0,...) in the main diagonal and (-1,-1,-1,...) in the subdiagonal.
%C A128174 Riordan array (1/(1-x^2), x) with inverse (1-x^2,x). - _Paul Barry_, Sep 10 2008
%C A128174 The position of 1's in this sequence is equivalent to A246705, and the position of 0's is equivalent to A246706. - _Bernard Schott_, Jun 05 2019
%H A128174 G. C. Greubel, <a href="/A128174/b128174.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A128174 A lower triangular matrix transform, (1, 0, 1, ...) in every column; n terms of (1, 0, 1, ...) in odd rows; n terms of (0, 1, 0, ...) in even rows.
%F A128174 T(n,k) = [k<=n]*(1+(-1)^(n-k))/2. - _Paul Barry_, Sep 10 2008
%F A128174 With offset n=1, k=0: Sum_{k=0..n} {T(n,k)*x^k} = A000035(n), A004526(n+1), A000975(n), A033113(n), A033114(n), A033115(n), A033116(n), A033117(n), A033118(n), A033119(n), A056830(n+1) for x=0,1,2,3,4,5,6,7,8,9,10 respectively. - _Philippe Deléham_, Oct 17 2011
%F A128174 T(n+1,1) = 1 - T(n,1); T(n+1,k) = T(n,k-1), 1 < k <= n+1. - _Reinhard Zumkeller_, Aug 01 2014
%e A128174 First few rows of the triangle are:
%e A128174   1;
%e A128174   0, 1;
%e A128174   1, 0, 1;
%e A128174   0, 1, 0, 1;
%e A128174   1, 0, 1, 0, 1; ...
%p A128174 A128174 := proc(n,k)
%p A128174     if k > n or k < 1 then
%p A128174         0;
%p A128174     else
%p A128174         modp(k+n+1,2) ;
%p A128174     end if;
%p A128174 end proc: # _R. J. Mathar_, Aug 06 2016
%t A128174 a128174[r_] := Table[If[EvenQ[n+k], 1, 0], {n, 1, r}, {k, 1, n}]
%t A128174 TableForm[a128174[5]] (* triangle *)
%t A128174 Flatten[a128174[10]] (* data *) (* _Hartmut F. W. Hoft_, Mar 15 2017 *)
%t A128174 Table[(1+(-1)^(n-k))/2, {n,1,12}, {k,1,n}]//Flatten (* _G. C. Greubel_, Sep 26 2017 *)
%o A128174 (Haskell)
%o A128174 a128174 n k = a128174_tabl !! (n-1) !! (k-1)
%o A128174 a128174_row n = a128174_tabl !! (n-1)
%o A128174 a128174_tabl = iterate (\xs@(x:_) -> (1 - x) : xs) [1]
%o A128174 -- _Reinhard Zumkeller_, Aug 01 2014
%o A128174 (PARI) for(n=1,12, for(k=1,n, print1((1+(-1)^(n-k))/2, ", "))) \\ _G. C. Greubel_, Sep 26 2017
%o A128174 (Magma) [[(1+(-1)^(n-k))/2: k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, Jun 05 2019
%o A128174 (Sage) [[(1+(-1)^(n-k))/2 for k in (1..n)] for n in (1..12)] # _G. C. Greubel_, Jun 05 2019
%Y A128174 Cf. A004526 (row sums).
%K A128174 nonn,easy,tabl
%O A128174 1,1
%A A128174 _Gary W. Adamson_, Feb 17 2007