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%I #9 Jan 05 2016 18:05:51
%S 1,1,2,7,45,578,15523,872933,101606058,24244705427,11781372347197,
%T 11604013148951290,23086334686919094283,92540040424223196349213,
%U 745956027717908362991989762,12075313247950868952015337447195,392133966660263237084551188748738021,25526209248562553823289966078580478182370,3328929359036770266032093183197212572084089475,869367378139090634989087434871564181449768656675093
%N G.f. A(x) satisfies: x = Sum_{n>=1} x^n * Product_{k=1..n} A(-k*x).
%C Compare to: x = Sum_{n>=1} x^n * F(-x)^n if F(x) = 1/(1-x).
%e 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 +...
%e where
%e 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) +...
%e The array of coefficients in [Product_{k=1..n} A(-k*x)] begins:
%e n=1: [1, -1, 2, -7, 45, -578, 15523, -872933, ...];
%e n=2: [1, -3, 12, -75, 851, -20052, 1030839, -113682051, ...];
%e n=3: [1, -6, 39, -354, 5504, -177612, 12901857, -2062308438, ...];
%e n=4: [1, -10, 95, -1150, 22376, -889420, 81529745, -16832853850, ...];
%e n=5: [1, -15, 195, -3000, 69751, -3229425, 351383145, -88131849450, ...];
%e n=6: [1, -21, 357, -6762, 182791, -9528099, 1183349175, -346858512000, ...];
%e n=7: [1, -28, 602, -13720, 423577, -24310860, 3353035806, -1117962684216, ...];
%e n=8: [1, -36, 954, -25704, 895065, -55714068, 8361452286, -3107960056872, ...]; ...
%e in which the antidiagonal sums yield [1,0,0,0,0,0,0,0,...].
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o 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]}
%o for(n=0,30, print1(a(n),", "))
%o (PARI) /* Quick print of terms 0..30 */
%o {A=[1]; for(i=1,30, A=concat(A,0);
%o A[#A]=(-1)^(#A)*Vec(sum(n=1,#A, prod(k=1,n,subst(Ser(A),x,-k*x))*x^n))[#A] );A}
%Y Cf. A266812.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 05 2016
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