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A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2*n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2*n+1)] are > 0.

%I A376277 #68 Mar 30 2025 06:31:45
%S A376277 1,2,5,13,35,98,287,883,2858,9708,34411,126337,476767,1836851,7185420,
%T A376277 28420613,113317776,454468077,1830556209,7397188271,29965426959,
%U A376277 121620119888,494365414071,2011965781648,8196475452837,33419092543257,136353532725534,556669705441210
%N A376277 The least increasing sequence starting with 1, such that the determinants of the Hankel matrices H1 = [a(0), a(1), ..., a(n); ...; a(n), a(n+1), ..., a(2*n)] and H2 = [a(1), a(2), ..., a(n+1); ...; a(n+1), a(n+2), ..., a(2*n+1)] are > 0.
%C A376277 A Stieltjes moment sequence by its definition.
%C A376277 The Hankel sequence transform gives {1, 1, 1, 1, 1, ...}.
%C A376277 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.
%F A376277 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.
%F A376277 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)).
%F A376277 (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).
%F A376277 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.
%F A376277 a(n) = Sum_{k=1..floor((n+1)/2)} (A034807(n+1, k) - A011973(n+1, k-4))*(-1)^(k+1)*a(n-k), for n >= 3.
%o A376277 (PARI)
%o A376277 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))}
%o A376277 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)}
%o A376277 (PARI)
%o A376277 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))))))
%o A376277 (PARI)
%o A376277 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)))
%Y A376277 Cf. A000108 (We obtain the Catalan numbers if we use "least positive sequence" in the definition instead of "least increasing").
%Y A376277 Cf. A375181 (Binomial transform).
%Y A376277 Cf. A034807, A011973.
%Y A376277 Cf. also A055877, A055878, A055879, A350349.
%K A376277 nonn,less
%O A376277 0,2
%A A376277 _Thomas Scheuerle_, Sep 23 2024