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A127872 Triangle formed by reading A039599 mod 2.

%I A127872 #14 Jan 23 2023 12:05:32
%S A127872 1,1,1,0,1,1,1,1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,1,1,
%T A127872 1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,
%U A127872 0,0,1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1
%N A127872 Triangle formed by reading A039599 mod 2.
%C A127872 Also triangle formed by reading triangles A061554, A106180, A110519, A124574, A124576, A126953, A127543 modulo 2.
%H A127872 G. C. Greubel, <a href="/A127872/b127872.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A127872 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A036987(n), A001316(n), A062878(n) for x=-1,0,1,2 respectively.
%F A127872 Sum_{k=0..n} T(n,k)*Fibonacci(2*k+1) = A050614(n), see A000045 and A001519. - _Philippe Deléham_, Aug 30 2007
%e A127872 Triangle begins:
%e A127872 1;
%e A127872 1, 1;
%e A127872 0, 1, 1;
%e A127872 1, 1, 1, 1;
%e A127872 0, 0, 0, 1, 1;
%e A127872 0, 0, 1, 1, 1, 1;
%e A127872 0, 1, 1, 0, 0, 1, 1;
%e A127872 1, 1, 1, 1, 1, 1, 1, 1;
%e A127872 0, 0, 0, 0, 0, 0, 0, 1, 1;
%e A127872 0, 0, 0, 0, 0, 0, 1, 1, 1, 1;
%e A127872 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1;
%e A127872 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;
%e A127872 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
%e A127872 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1;
%e A127872 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1;
%e A127872 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; ...
%t A127872 T[0, 0] := 1; T[n_, k_] := Binomial[2*n - 1, n - k] - Binomial[2*n - 1, n - k - 2]; Table[Mod[T[n, k], 2], {n,0,10}, {k,0,n}] // Flatten (* _G. C. Greubel_, Apr 18 2017 *)
%Y A127872 Cf. A061554, A106180, A110519, A124574, A124576, A126953, A127543.
%Y A127872 Cf. A000007, A036987, A001316, A062878, A050614, A000045, A001519
%K A127872 nonn,tabl
%O A127872 0,1
%A A127872 _Philippe Deléham_, Apr 05 2007