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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 45*x^4 + 578*x^5 + 15523*x^6 + 872933*x^7 + 101606058*x^8 + 24244705427*x^9 + 11781372347197*x^10 +...
where
x = x*A(-x) + x^2*A(-x)*A(-2*x) + x^3*A(-x)*A(-2*x)*A(-3*x) + x^4*A(-x)*A(-2*x)*A(-3*x)*A(-4*x) + x^5*A(-x)*A(-2*x)*A(-3*x)*A(-4*x)*A(-5*x) +...
The array of coefficients in [Product_{k=1..n} A(-k*x)] begins:
n=1: [1, -1, 2, -7, 45, -578, 15523, -872933, ...];
n=2: [1, -3, 12, -75, 851, -20052, 1030839, -113682051, ...];
n=3: [1, -6, 39, -354, 5504, -177612, 12901857, -2062308438, ...];
n=4: [1, -10, 95, -1150, 22376, -889420, 81529745, -16832853850, ...];
n=5: [1, -15, 195, -3000, 69751, -3229425, 351383145, -88131849450, ...];
n=6: [1, -21, 357, -6762, 182791, -9528099, 1183349175, -346858512000, ...];
n=7: [1, -28, 602, -13720, 423577, -24310860, 3353035806, -1117962684216, ...];
n=8: [1, -36, 954, -25704, 895065, -55714068, 8361452286, -3107960056872, ...]; ...
in which the antidiagonal sums yield [1,0,0,0,0,0,0,0,...].
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A]=(-1)^(#A)*Vec(sum(m=1, #A, prod(k=1, m, subst(Ser(A), x, -k*x))*x^m))[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Quick print of terms 0..30 */
{A=[1]; for(i=1, 30, A=concat(A, 0);
A[#A]=(-1)^(#A)*Vec(sum(n=1, #A, prod(k=1, n, subst(Ser(A), x, -k*x))*x^n))[#A] ); A}
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