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Triangle T(n,k), read by rows: T(n,k) is the numerator of (1-2^(n-k+1))/(1-2^(k+1)).
2

%I #20 Sep 08 2022 08:46:05

%S 1,3,1,7,1,1,15,7,3,1,31,5,1,1,1,63,31,15,7,3,1,127,21,31,1,7,1,1,255,

%T 127,9,31,15,1,3,1,511,85,127,21,1,5,7,1,1,1023,511,255,127,63,31,15,

%U 7,3,1,2047,341,73,17,127,1,31,1,1,1,1,4095,2047,1023

%N Triangle T(n,k), read by rows: T(n,k) is the numerator of (1-2^(n-k+1))/(1-2^(k+1)).

%C The denominators are given in A228035.

%C The first column is A000225, the second column is A213243, and the third column is A213245.

%H Vincenzo Librandi, <a href="/A227984/b227984.txt">Rows n = 0..100, flattened</a>

%e Triangle begins:

%e 1;

%e 3, 1;

%e 7, 1, 1;

%e 15, 7, 3, 1;

%e 31, 5, 1, 1, 1;

%e 63, 31, 15, 7, 3, 1;

%e 127, 21, 31, 1, 7, 1, 1;

%e 255, 127, 9, 31, 15, 1, 3, 1;

%e 511, 85, 127, 21, 1, 5, 7, 1, 1;

%e 1023, 511, 255, 127, 63, 31, 15, 7, 3, 1;

%e 2047, 341, 73, 17, 127, 1, 31, 1, 1, 1, 1; etc.

%t a[n_, k_] := Numerator[(1 - 2^(n - k + 1))/(1 - 2^(k + 1))];

%t Table[a[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

%o (Magma) [Numerator((1-2^(n-k+1))/(1-2^(k+1))): k in [0..n], n in [0..11]];

%Y Cf. A000225, A213243, A213245, A228035.

%K nonn,tabl,frac

%O 0,2

%A _Vincenzo Librandi_, Aug 12 2013