OFFSET
0,3
LINKS
Wikipedia, Moment problem.
FORMULA
Let s(n) be the Hankel transform of this sequence. Starting: 1, 1, 1, 2, 8, 2400, 1440000,... .
s(n) = Product_{m=0..n-2} Product_{k=1..m} a(A053645(2*k))*a(A053645(2*k+1)). This formula is particular interesting here, because if we expand the Hankel determinants based on A380645: s(0) = 1, s(1) = a(0), s(2) = a(0)*a(2) - a(1)^2, ..., then we will be able to obtain A380645 directly from a system of equations. This means the product formula for the Hankel determinant is also definition of this sequence.
The Hankel transforms of this sequence starting with offset 0 and starting with offset 1 are both strictly positive, this means this sequence can be considered as a sequence of moments in the Stieltjes moment problem: a(n) = Integral_{x>=0} x^n*V(n, x) dx. It is conjectured that the measure V(n, x) is related to some kind of interesting distribution which shows self similarity over intervals of powers of two.
EXAMPLE
1, 1, 1, 2, 1, 2, 5, 15, 1, 2, 5, 15, 52, 201, 857, 4370, 1, 2, 5, 15, 52, 201, 857, 4370, 34365, 478287, 9095996, 189526537, 4036216585, 87129122290, 2478683501397, 2450240534552191, 1, 2, 5, 15, ...
This also forms a fractal sequence as it is invariant under the removal of each term's first occurrence:
, 1, 1, , 1, 2, , , 1, 2, 5, 15, , , , , 1, 2, 5, 15, 52, 52, 201, 857, ...
We insert this fractal sequence into the Stilties continued fractions and expand:
1/(1 - 1*x/(1 - 1*x/(1 - 1*x/(1 - 2*x/(1 - 1*x/(1 - 2*x/(1 - 5*x/(...)))))))) =
1 + 1*x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 201*x^6 + 857*x^7 + 4070*x^8 + 21765*x^9 + ... .
PROG
(PARI)
cf(v) = {my(f = O(x)); for(k=1, #v, f=1/(1-v[#v+1-k]*x*f)); Vec(f+O(x^(#v)))}
vectorA(numiter) = {my(v = [1]); for(k=0, numiter, v=concat(v, cf(v))); v[1+#v/2..#v]}
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Thomas Scheuerle, Feb 06 2025
STATUS
approved