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

0,8

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

Table of n, a(n) for n=0..80.

OEIS Wiki, Autosequence

Wikipedia, Autosuite de nombres (in French).

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

Crossrefs

Cf. A001045, A007318, A069834, A137688.

Sequence in context: A088204 A100375 A353063 * A066975 A355879 A291634

Adjacent sequences: A274610 A274611 A274612 * A274614 A274615 A274616

Keyword

easy,nonn,tabl

Author

Jean-François Alcover and Paul Curtz, Jul 07 2016

Status

approved