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 \( f(x) \) 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 \( V(x)=[1,x,x^2,x^2,x^3,...] \)).
Then the dot-product \( V(x)*M=Y \) gives a rowvector Y whose entries evaluate to geometric series such that
\( V(x)*M = 1/(1-x) * [1,x^2,x^4,x^6,...] = 1/(1-x)*V(x^2) \).
Clearly this can be iterated:
\( V(x^2)*M = 1/(1-x^2) * [1,x^4,x^8,x^{12},...] = 1/(1-x^2)*V(x^4) \)
and expressed with the power of M
\( V(x)*M = 1/(1-x) * V(x^{2^1}) \).
\( V(x)*M^2 = 1/(1-x)*1/(1-x^2) * V(x^{2^2}) \).
...
\( V(x)*M^h = 1/(1-x)*1/(1-x^2)*...*1/(1-x^{2^h}) * V(x^{2^{h+1}}) \) .
In the limit to infinite powers of M this gives for the first column in the result the scalar value \( ... \)
\( y = w(x) = 1 / (1-x) / (1-x ^ 2) / (1 - x ^ 4 ) \ldots \)
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:
\( y = w(x) = 1 + 1x + 2x^2 + 2x^3 + 4x^4+4x^5+... \)
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 \( f(x) \) 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 \( V(x)=[1,x,x^2,x^2,x^3,...] \)).
Then the dot-product \( V(x)*M=Y \) gives a rowvector Y whose entries evaluate to geometric series such that
\( V(x)*M = 1/(1-x) * [1,x^2,x^4,x^6,...] = 1/(1-x)*V(x^2) \).
Clearly this can be iterated:
\( V(x^2)*M = 1/(1-x^2) * [1,x^4,x^8,x^{12},...] = 1/(1-x^2)*V(x^4) \)
and expressed with the power of M
\( V(x)*M = 1/(1-x) * V(x^{2^1}) \).
\( V(x)*M^2 = 1/(1-x)*1/(1-x^2) * V(x^{2^2}) \).
...
\( V(x)*M^h = 1/(1-x)*1/(1-x^2)*...*1/(1-x^{2^h}) * V(x^{2^{h+1}}) \) .
In the limit to infinite powers of M this gives for the first column in the result the scalar value \( ... \)
\( y = w(x) = 1 / (1-x) / (1-x ^ 2) / (1 - x ^ 4 ) \ldots \)
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:
\( y = w(x) = 1 + 1x + 2x^2 + 2x^3 + 4x^4+4x^5+... \)
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