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.

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, 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, 165, 33, 13, 1, 1, 21, 63, 165, 55, 39, 7, 1, 1, 9, 105, 231, 495, 143, 91, 15, 1

Offset: 0

Author

Jean-François Alcover and Paul Curtz, Jul 07 2016

Keywords

• easy
• nonn
• tabl

Comments

Array of fractions begins:

1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, ...

0, 1/2, 1/2, 3/8, 1/4, 5/32, 3/32, 7/128, ...

0, 0, 1/4, 3/8, 3/8, 5/16, 15/64, 21/128, ...

0, 0, 0, 1/8, 1/4, 5/16, 5/16, 35/128, ...

0, 0, 0, 0, 1/16, 5/32, 15/64, 35/128, ...

0, 0, 0, 0, 0, 1/32, 3/32, 21/128, ...

0, 0, 0, 0, 0, 0, 1/64, 7/128, ...

...

Given the symmetry T(n,k) = T(k-n,k) in the upper triangle, rows and upper diagonals are identical.

The first row, which is also the main diagonal, is 1/2^k.

The second row is Oresme numbers k/2^k.

The third row is (k(k-1)/2!)/2^k (see A069834).

The fourth row is (k(k-1)(k-2)/3!)/2^k.

The sum of any column is always 1.

Omitting the zeros, the columns are fractions A007318/A137688.

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).

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.

Links

• OEIS Wiki, Autosequence

Crossrefs

Cf. A001045, A007318, A069834, A137688.

Programs

Mathematica

T[n_, k_] := Binomial[k, n]/2^k;
Table[T[n - k, k] // Numerator, {n, 0, 16}, {k, Floor[(n + 1)/2], n}] // Flatten