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A133080 Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.

%I A133080 #24 Jan 13 2022 02:26:48
%S A133080 1,1,1,0,0,1,0,0,1,1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,
%T A133080 1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U A133080 0,0,0,0,0,0,0,0,1,1
%N A133080 Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
%C A133080 A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, ...).
%C A133080 Inverse of A133080: subdiagonal changes to (-1, 0, -1, 0, -1, ...); main diagonal unchanged.
%C A133080 A133080^(-1) * [1,2,3,...] = A093178: (1, 1, 3, 1, 5, 1, 7, 1, 9, ...).
%C A133080 In A133081, diagonal terms are switched with subdiagonal terms.
%H A133080 G. C. Greubel, <a href="/A133080/b133080.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A133080 Infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (1,0,1,0,1,...) in the subdiagonal.
%F A133080 Odd rows, (n-1) zeros followed by "1". Even rows, (n-2) zeros followed by "1, 1".
%F A133080 T(n,n)=1. T(n,k)=0 if 1 <= k < n-1. T(n,n-1)=1 if n even. T(n,n-1)=0 if n odd. - _R. J. Mathar_, Feb 14 2015
%e A133080 First few rows of the triangle are:
%e A133080   1;
%e A133080   1, 1;
%e A133080   0, 0, 1;
%e A133080   0, 0, 1, 1;
%e A133080   0, 0, 0, 0, 1;
%e A133080   0, 0, 0, 0, 1, 1;
%e A133080   0, 0, 0, 0, 0, 0, 1;
%e A133080   ...
%p A133080 A133080 := proc(n,k)
%p A133080     if n = k then
%p A133080         1;
%p A133080     elif  k=n-1 and type(n,even) then
%p A133080         1;
%p A133080     else
%p A133080         0 ;
%p A133080     end if;
%p A133080 end proc: # _R. J. Mathar_, Jun 20 2015
%t A133080 T[n_, k_] := If[k == n, 1, If[k == n - 1, (1 + (-1)^n)/2 , 0]];
%t A133080 Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* _G. C. Greubel_, Oct 21 2017 *)
%o A133080 (PARI) T(n, k) = if (k==n, 1, if (k == (n-1), 1 - (n % 2), 0)); \\ _Michel Marcus_, Feb 13 2014
%o A133080 (PARI) firstrows(n) = {my(res = vector(binomial(n + 1, 2)), t=0); for(i=1, n, t+=i; res[t] = 1; if(i%2==0, res[t-1]=1)) ;res} \\ _David A. Corneth_, Oct 21 2017
%Y A133080 Cf. A000034 (row sums), A114753, A093178, A133081.
%K A133080 nonn,easy,tabl
%O A133080 1,1
%A A133080 _Gary W. Adamson_, Sep 08 2007