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A188109 Triangle T(n,m), [x*A(x)]^m=sum(n>=m T(n,m)*x^n), where A(x) satisfies x*A(x)^2= -(2*x*A(x)+sqrt(1-4*x*A(x))-1)/(4*x*A(x)+sqrt(1-4*x*A(x))-1).

%I A188109 #20 Jun 02 2025 03:59:50
%S A188109 1,3,1,19,6,1,152,47,9,1,1367,418,84,12,1,13195,4007,825,130,15,1,
%T A188109 133556,40368,8433,1400,185,18,1,1398696,421332,88872,15239,2170,249,
%U A188109 21,1,15029311,4515706,959080,168112,25100,3162,322,24,1,164764985,49405895,10547361,1878462,289788,38772,4403,404,27,1
%N A188109 Triangle T(n,m), [x*A(x)]^m=sum(n>=m T(n,m)*x^n), where A(x) satisfies x*A(x)^2= -(2*x*A(x)+sqrt(1-4*x*A(x))-1)/(4*x*A(x)+sqrt(1-4*x*A(x))-1).
%H A188109 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 A188109 T(n,m) = m/n*A035324(2*n-m,n).
%e A188109 1;
%e A188109 3, 1;
%e A188109 19, 6, 1;
%e A188109 152, 47, 9, 1;
%e A188109 1367, 418, 84, 12, 1;
%e A188109 13195, 4007, 825, 130, 15, 1;
%e A188109 133556, 40368, 8433, 1400, 185, 18, 1;
%e A188109 1398696, 421332, 88872, 15239, 2170, 249, 21, 1
%t A188109 (* S = A035324 *)
%t A188109 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 A188109 T[n_, m_] := m/n S[2n-m, n];
%t A188109 Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 16 2019 *)
%Y A188109 Cf. A035324
%K A188109 nonn,tabl
%O A188109 1,2
%A A188109 _Vladimir Kruchinin_, Mar 21 2011