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A188110 Triangle T(n,m), [x*A(x)]^m=sum(n>=m T(n,m)*x^n), where A(x) satisfies x*A(x)^3= -(2*x*A(x)^2+sqrt(1-4*x*A(x)^2)-1)/(4*x*A(x)^2+sqrt(1-4*x*A(x)^2)-1).

%I A188110 #21 Jun 02 2025 03:59:57
%S A188110 1,3,1,28,6,1,350,65,9,1,5020,868,111,12,1,78023,12924,1581,166,15,1,
%T A188110 1278340,205766,24468,2516,230,18,1,21740636,3428438,399735,40489,
%U A188110 3700,303,21,1,380161308,59034600,6784186,679460,61905,5160,385,24,1,6792111260,1042169972,118444293,11759612,1067738,89715,6923,476,27,1
%N A188110 Triangle T(n,m), [x*A(x)]^m=sum(n>=m T(n,m)*x^n), where A(x) satisfies x*A(x)^3= -(2*x*A(x)^2+sqrt(1-4*x*A(x)^2)-1)/(4*x*A(x)^2+sqrt(1-4*x*A(x)^2)-1).
%H A188110 Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A188110 T(n,m) = m/(2*n-m)*A035324(3*n-2*m,2*n-m).
%t A188110 (* S = A035324 *)
%t A188110 S[n_, m_] /; n >= m >= 1 := S[n, m] = 2(2(n-1) + m)(S[n - 1, m]/n) + m (S[n - 1, m - 1]/n); S[n_, m_] /; n < m = 0; S[n_, 0] = 0; S[1, 1] = 1;
%t A188110 T[n_, m_] := m/(2n-m) S[3n - 2m, 2n - m];
%t A188110 Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 16 2019 *)
%Y A188110 Cf. A035324.
%K A188110 nonn,tabl
%O A188110 1,2
%A A188110 _Vladimir Kruchinin_, Mar 21 2011