User:Jf.alcover/sandbox

Definition
"Autosequence" is a neology once coined by Paul Curtz, well-known contributor to the OEIS, to give a name to the sequences that are identical to their inverse binomial transform, up to alternating signs.

This means that for such an autosequence a(n), the identity:

(-1)^n a(n) = sum_{k=0..n} (-1)^(n-k) binomial(n,k) a(k)

holds for all positive n.

Two kinds of autosequences may be distinguished, according to the nature of the diagonal of the array of successive differences.

Autosequence of the first kind
An autosequence belongs to the first kind if and only if the main diagonal terms of its array of successive differences are all zeroes.

Example
Fibonacci(n) n>=0 (Cf. ) is an autosequence of the first kind, its successive differences being:  0,  1,   1,   2,   3,   5,   8,  ...

1,  0,   1,   1,   2,   3,   5,  ...

-1, 1,   0,   1,   1,   2,   3,  ...

2, -1,   1,   0,   1,   1,   2,  ...

-3, 2,  -1,   1,   0,   1,   1,  ...

5, -3,   2,  -1,   1,   0,   1,  ...

-8, 5,  -3,   2,  -1,   1,   0,  ...

...

Autosequence of the second kind
An autosequence belongs to the second kind if and only if the main diagonal of its array of successive differences is twice the first upper subdiagonal.

Example
Fibonacci(n)/n n>=1 (Cf. ) is an autosequence of the second kind, its successive differences being:  1,    1/2,   2/3,    3/4,       1, ...

-1/2, 1/6,  1/12,    1/4,     1/3, ...

2/3, -1/12,  1/6,   1/12,    4/21, ...

-3/4, 1/4, -1/12,   3/28,    3/56, ...

1,   -1/3,  4/21,  -3/56,  11/126, ...

...

Notes and references

 * Paul Barry, 2005, A Catalan Transform and Related Transformations on Integer Sequences


 * Helmut Prodinger, 1992, Some information about the Binomial transform


 * E. W. Weisstein, Binomial Transform


 * For more information, see Autosuite de nombres (in French)