A380645 - OEIS

%I #26 Feb 07 2025 15:58:12

%S 1,1,2,5,15,52,201,857,4370,34365,478287,9095996,189526537,4036216585,

%T 87129122290,2478683501397,2450240534552191,12482183328151728692,

%U 65634092872761268943625,345370818796643845031835465,1817414952852912380501431441282

%N 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Moment_problem">Moment problem</a>.

%F Let s(n) be the Hankel transform of this sequence. Starting: 1, 1, 1, 2, 8, 2400, 1440000,... .

%F 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.

%F 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.

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

%e 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, ...

%e This also forms a fractal sequence as it is invariant under the removal of each term's first occurrence:

%e  , 1, 1,  , 1, 2,  ,  , 1, 2, 5, 15,  ,  ,  ,  , 1, 2, 5, 15, 52, 52, 201, 857, ...

%e We insert this fractal sequence into the Stilties continued fractions and expand:

%e   1/(1 - 1*x/(1 - 1*x/(1 - 1*x/(1 - 2*x/(1 - 1*x/(1 - 2*x/(1 - 5*x/(...)))))))) =

%e   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 + ... .

%o (PARI)

%o cf(v) = {my(f = O(x)); for(k=1, #v, f=1/(1-v[#v+1-k]*x*f)); Vec(f+O(x^(#v)))}

%o vectorA(numiter) = {my(v = [1]); for(k=0, numiter, v=concat(v, cf(v))); v[1+#v/2..#v]}

%Y Cf. A053645.

%K nonn,new

%O 0,3

%A _Thomas Scheuerle_, Feb 06 2025