A127872 - OEIS

%I #14 Jan 23 2023 12:05:32

%S 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 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 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 Triangle formed by reading A039599 mod 2.

%C Also triangle formed by reading triangles A061554, A106180, A110519, A124574, A124576, A126953, A127543 modulo 2.

%H G. C. Greubel, <a href="/A127872/b127872.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F 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 Sum_{k=0..n} T(n,k)*Fibonacci(2*k+1) = A050614(n), see A000045 and A001519. - _Philippe Deléham_, Aug 30 2007

%e Triangle begins:

%e 1;

%e 1, 1;

%e 0, 1, 1;

%e 1, 1, 1, 1;

%e 0, 0, 0, 1, 1;

%e 0, 0, 1, 1, 1, 1;

%e 0, 1, 1, 0, 0, 1, 1;

%e 1, 1, 1, 1, 1, 1, 1, 1;

%e 0, 0, 0, 0, 0, 0, 0, 1, 1;

%e 0, 0, 0, 0, 0, 0, 1, 1, 1, 1;

%e 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1;

%e 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1;

%e 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1;

%e 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1;

%e 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1;

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; ...

%t 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 Cf. A061554, A106180, A110519, A124574, A124576, A126953, A127543.

%Y Cf. A000007, A036987, A001316, A062878, A050614, A000045, A001519

%K nonn,tabl

%O 0,1

%A _Philippe Deléham_, Apr 05 2007