OFFSET
0,2
COMMENTS
A Stieltjes moment sequence by its definition.
The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.
The definition causes that the Hankel sequence transform starting with the second term of this sequence becomes {2, 1, 1, 1, ...}. This single exceptional 2 causes high complexity in the generating function and makes a nice combinatorial interpretation less likely, therefore the keyword "less" was considered.
FORMULA
G.f.: 1/(1-2*x/(1-(1/2)*x/(1-(1/2)*x/(1-2*x/(1-C(x)*x))))), C(x) is the generating function of the Catalan numbers.
G.f.: (1 - sqrt(1 - 4*x)*(-1 + x) - 5*x + 2*x^2)/(1 - 7*x + 11*x^2 + sqrt(1 - 4*x)*(1 - 3*x + x^2)).
(sqrt((x - 4)/x) + 2*x*(13 + (x - 7)*x) - 9)/(2*((x - 4)*(x - 3)*(x - 2)*x - 1)) = Sum_{k>=0} a(k)/x^(k+1).
a(n) = Sum_{k=1..floor((n+1)/2)} (binomial(n-k+1, k) + binomial(n-k, k-1) - binomial(n-k-3, k-4))*(-1)^(k+1)*a(n-k)), for n >= 3.
PROG
(PARI)
hankelok(s) = {my(m1=floor((#s+1)/2)); my(m2=floor(#s/2)); my(h1=matrix(m1, m1, x, y, s[x+y-1])); my(h2=matrix(m2, m2, x, y, s[x+y])); return((matdet(h1) > 0) && (matdet(h2) > 0))}
a(max_n) = {my(s=[1, 2], k=3); while(#s < max_n, while(hankelok(concat(s, [k]))==0, k=k+1); s=concat(s, [k])); return(s)}
(PARI)
my(N=30, x='x+O('x^N)); Vec(1/(1-2*x/(1-(1/2)*x/(1-(1/2)*x/(1-2*x/(1-((1-sqrt(1-4*x))/(2*x))*x))))))
(PARI)
a(n) = if(n<3, [1, 2, 5][n+1], sum(k=1, floor((n+1)/2), (binomial(n-k+1, k)+binomial(n-k, k-1)-binomial(n-k-3, k-4))*(-1)^(k+1)*a(n-k)))
CROSSREFS
KEYWORD
nonn,less,new
AUTHOR
Thomas Scheuerle, Sep 23 2024
STATUS
approved