0,4

The inverse binomial transform yields the sequence (-1)^(n+1)*a(n). This property is inherited from the A000045 and A113405 sequences, which have the same property individually. The same sign flipping behavior under inverse binomial transform is found in A001045 and for the sequence with two zeros followed by A000975.

This is often, but not here, related to the recurrences a(n)=2a(n-1)+a(n-2)-2a(n-3) associated with denominators 1-2x-x^2+2x^3=(x-1)(2x-1)(x+1) in the o.g.f., which transform into the similar -(x-1)(2x+1)/(1+x)^4 under the inverse binomial transform, see A137241.

O.g.f.: x(1-2x-3x^4+x^2)/((1-x-x^2)(2x-1)(1+x)(x^2-x+1)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 10 2008

a(n)= -A128834(n)/3+2^n/9+A000045(n)-(-1)^n/9. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 10 2008

a(n) and the repeated differences in the followup rows are:

0, 1, 1, 3, 5, 9, 15,..

1, 0, 2, 2, 4, 6, 12,..

-1, 2, 0, 2, 2, 6, 10,..

3, -2, 2, 0, 4, 4, 10,..

-5, 4, -2, 4, 0, 6, 6,..

9, -6, 6, -4, 6, 0, 12,..

-15, 12, -10, 10, -6, -12, 0,..

The main diagonal contains zeros.

Sequence in context: A018436 A018298 A017913 this_sequence A027154 A052007 A117480

Adjacent sequences: A140425 A140426 A140427 this_sequence A140429 A140430 A140431

nonn

Paul Curtz (bpcrtz(AT)free.fr), Jun 19 2008

Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 10 2008