I've got the generating rule in terms of a matrix. Assume an (ideally infinite sized) matrix with a simple generating-scheme, whose top-left segments is

Then the sequence A018819 occurs by taking M to infinite powers. Because the diagonal is zero except of the top left element M is nilpotent and the powers of M tend to become a column-vector in the first column. See for instance M^2:

and a higher power M^6:

Usually I try such approaches to consider diagonalization or Jordan-forms et al - perhaps for a suggestive pattern, for instance, A is an eigensequence for M because

However I did not yet find more interesting things in this manner.

Interestingly the sequence generated by Jay's function for real valued approximations to the elements of A has a vanishing binomial-transform: if that sequence of coefficients is premultiplied by the matrix P^-1 (the inverse of the lower triangular Pascalmatrix) then the transformed coefficients vanish quickly: (from left to right Jay's coefficients, the binomial-transforms, the original coefficients, and the binomial-transforms of the original coefficients):

One more observation which might be interesting.

Consider the dotproduct of a Vandermondevector V(x) with M (where V(x) is a vector of consecutive powers of x such that ).

Then the dot-product gives a rowvector Y whose entries evaluate to geometric series such that

.

Clearly this can be iterated:

and expressed with the power of M

.

.

...

.

In the limit to infinite powers of M this gives for the first column in the result the scalar value

and all others columns tend to zero. We might say, that in the above notation w(x) is the generating function for the sequence A.

update: I changed the name of the function to not to interfer with Jay's function f(x) which interpolates the sequence A by real values of f(x)

The *value* y, on the other hand, is then the evaluation of the power series whose coefficients are the terms of the original sequence at x:

which seems to be convergent for |x|<1.

I'm fiddling a bit more with it but do not yet expect much exciting news...

Gottfried

Then the sequence A018819 occurs by taking M to infinite powers. Because the diagonal is zero except of the top left element M is nilpotent and the powers of M tend to become a column-vector in the first column. See for instance M^2:

and a higher power M^6:

Usually I try such approaches to consider diagonalization or Jordan-forms et al - perhaps for a suggestive pattern, for instance, A is an eigensequence for M because

However I did not yet find more interesting things in this manner.

Interestingly the sequence generated by Jay's function for real valued approximations to the elements of A has a vanishing binomial-transform: if that sequence of coefficients is premultiplied by the matrix P^-1 (the inverse of the lower triangular Pascalmatrix) then the transformed coefficients vanish quickly: (from left to right Jay's coefficients, the binomial-transforms, the original coefficients, and the binomial-transforms of the original coefficients):

One more observation which might be interesting.

Consider the dotproduct of a Vandermondevector V(x) with M (where V(x) is a vector of consecutive powers of x such that ).

Then the dot-product gives a rowvector Y whose entries evaluate to geometric series such that

.

Clearly this can be iterated:

and expressed with the power of M

.

.

...

.

In the limit to infinite powers of M this gives for the first column in the result the scalar value

and all others columns tend to zero. We might say, that in the above notation w(x) is the generating function for the sequence A.

update: I changed the name of the function to not to interfer with Jay's function f(x) which interpolates the sequence A by real values of f(x)

The *value* y, on the other hand, is then the evaluation of the power series whose coefficients are the terms of the original sequence at x:

which seems to be convergent for |x|<1.

I'm fiddling a bit more with it but do not yet expect much exciting news...

Gottfried

Gottfried Helms, Kassel