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A348901 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(2*x)).

%I A348901 #23 Jul 10 2025 06:35:21
%S A348901 1,1,5,49,893,30649,2030213,264198625,68180168717,35046644401609,
%T A348901 35958357173552597,73714882938928013809,302083844634245306686685,
%U A348901 2475275541582550287356775001,40559867144321249927245807932197,1329146863668196853655964629931680001
%N A348901 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(2*x)).
%C A348901 Counts lower triangular (0,1) matrices with 1's on the diagonal which cannot be decomposed in a nontrivial block diagonal fashion. For example, the third time is 5, counting the matrices [100,110,111], [100,110,011], [100,010,111],  [100,110,101], [100,010,101]. There are 3 other 3x3 lower triangular (0,1) matrices with 1's on the diagonal; those others have block decompositions. - _David Speyer_, Jul 09 2025
%H A348901 Michael De Vlieger, <a href="/A348901/b348901.txt">Table of n, a(n) for n = 0..81</a>
%H A348901 Johann Cigler, <a href="https://arxiv.org/abs/2407.05768">Hankel determinants of backward shifts of powers of q</a>, arXiv:2407.05768 [math.CO], 2024. See p. 6.
%H A348901 Mare et al., <a href="https://mathoverflow.net/q/497423/297">Counting lower triangular 0-1-matrices with connected Coxeter permutation</a>, MathOverflow, 2025.
%F A348901 a(0) = 1; a(n) = -a(n-1) + Sum_{k=0..n-1} 2^(k+1) * a(k) * a(n-k-1).
%F A348901 a(n) ~ 2^(n*(n+1)/2). - _Vaclav Kotesovec_, Nov 03 2021
%F A348901 G.f. A(x) satisfies 1/(1 - x*A(x)) = Sum_{n>=0} 2^(n(n-1)/2) * x^n. - _David Speyer_, Jul 09 2025
%t A348901 nmax = 15; A[_] = 0; Do[A[x_] = 1/(1 + x - 2 x A[2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A348901 a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[2^(k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
%Y A348901 Cf. A001003, A015083, A348188, A348902.
%K A348901 nonn
%O A348901 0,3
%A A348901 _Ilya Gutkovskiy_, Nov 03 2021