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A274613 Array T(n,k) = numerator of binomial(k,n)/2^k read by antidiagonals omitting the zeros (upper triangle), a sequence related to Jacobsthal numbers.

%I A274613 #30 Oct 18 2024 14:36:56
%S A274613 1,1,1,1,1,1,1,3,1,3,1,1,1,3,5,1,1,5,3,1,1,5,15,7,1,5,5,21,1,1,1,15,
%T A274613 35,7,9,1,3,35,7,9,5,1,1,21,35,21,45,11,1,7,7,63,15,55,3,1,1,7,63,105,
%U A274613 165,33,13,1,1,21,63,165,55,39,7,1,1,9,105,231,495,143,91,15,1
%N A274613 Array T(n,k) = numerator of binomial(k,n)/2^k read by antidiagonals omitting the zeros (upper triangle), a sequence related to Jacobsthal numbers.
%C A274613 Array of fractions begins:
%C A274613 1,   1/2,   1/4,   1/8,  1/16,  1/32,    1/64,    1/128, ...
%C A274613 0,   1/2,   1/2,   3/8,   1/4,  5/32,    3/32,    7/128, ...
%C A274613 0,     0,   1/4,   3/8,   3/8,  5/16,   15/64,   21/128, ...
%C A274613 0,     0,     0,   1/8,   1/4,  5/16,    5/16,   35/128, ...
%C A274613 0,     0,     0,     0,  1/16,  5/32,   15/64,   35/128, ...
%C A274613 0,     0,     0,     0,     0,  1/32,    3/32,   21/128, ...
%C A274613 0,     0,     0,     0,     0,     0,    1/64,    7/128, ...
%C A274613 ...
%C A274613 Given the symmetry T(n,k) = T(k-n,k) in the upper triangle, rows and upper diagonals are identical.
%C A274613 The first row, which is also the main diagonal, is 1/2^k.
%C A274613 The second row is Oresme numbers k/2^k.
%C A274613 The third row is (k(k-1)/2!)/2^k (see A069834).
%C A274613 The fourth row is (k(k-1)(k-2)/3!)/2^k.
%C A274613 The sum of any column is always 1.
%C A274613 Omitting the zeros, the columns are fractions A007318/A137688.
%C A274613 The sum of the n-th antidiagonal is A001045(n+1)/2^n; the numerators of these sums are the positive Jacobsthal numbers 1, 1, 3, 5, 11, 21, 43, 85, ... (see A001045).
%C A274613 It can also be observed that every row is an "autosequence", that is a sequence which is identical to its inverse binomial transform, except for signs.
%H A274613 OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%t A274613 T[n_, k_] := Binomial[k, n]/2^k;
%t A274613 Table[T[n - k, k] // Numerator, {n, 0, 16}, {k, Floor[(n + 1)/2], n}] // Flatten
%Y A274613 Cf. A001045, A007318, A069834, A137688.
%K A274613 easy,nonn,tabl
%O A274613 0,8
%A A274613 _Jean-François Alcover_ and _Paul Curtz_, Jul 07 2016