A380645 The expansion of the Stieltjes continued fraction 1/(1 - x/(1 - a(A053645(0))*x/(1 - a(A053645(1))*x/(1 - a(A053645(2))*x/...)))) gives the sequence itself.

1, 1, 2, 5, 15, 52, 201, 857, 4370, 34365, 478287, 9095996, 189526537, 4036216585, 87129122290, 2478683501397, 2450240534552191, 12482183328151728692, 65634092872761268943625, 345370818796643845031835465, 1817414952852912380501431441282

Offset: 0

Author

Thomas Scheuerle, Feb 06 2025

Keywords

• nonn

Examples

			The sequence 1, a(A053645(0)), a(A053645(1)), ... begins:
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 + ... .
		

Links

• Wikipedia, Moment problem.

Crossrefs

Cf. A053645.

Programs

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]}

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.