|
%I #26 May 07 2017 12:07:08
%S 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 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 165,33,13,1,1,21,63,165,55,39,7,1,1,9,105,231,495,143,91,15,1
%N 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 Array of fractions begins:
%C 1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, ...
%C 0, 1/2, 1/2, 3/8, 1/4, 5/32, 3/32, 7/128, ...
%C 0, 0, 1/4, 3/8, 3/8, 5/16, 15/64, 21/128, ...
%C 0, 0, 0, 1/8, 1/4, 5/16, 5/16, 35/128, ...
%C 0, 0, 0, 0, 1/16, 5/32, 15/64, 35/128, ...
%C 0, 0, 0, 0, 0, 1/32, 3/32, 21/128, ...
%C 0, 0, 0, 0, 0, 0, 1/64, 7/128, ...
%C ...
%C Given the symmetry T(n,k) = T(k-n,k) in the upper triangle, rows and upper diagonals are identical.
%C The first row, which is also the main diagonal, is 1/2^k.
%C The second row is Oresme numbers k/2^k.
%C The third row is (k(k-1)/2!)/2^k (see A069834).
%C The fourth row is (k(k-1)(k-2)/3!)/2^k.
%C The sum of any column is always 1.
%C Omitting the zeros, the columns are fractions A007318/A137688.
%C 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 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 OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%H Wikipedia, <a href="https://fr.wikipedia.org/wiki/Autosuite_de_nombres">Autosuite de nombres</a> (in French).
%t T[n_, k_] := Binomial[k, n]/2^k;
%t Table[T[n - k, k] // Numerator, {n, 0, 16}, {k, Floor[(n + 1)/2], n}] // Flatten
%Y Cf. A001045, A007318, A069834, A137688.
%K easy,nonn,tabl
%O 0,8
%A _Jean-François Alcover_ and _Paul Curtz_, Jul 07 2016
|
