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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 231*x^4 + 3333*x^5 + 58167*x^6 + 1175877*x^7 + 26827623*x^8 + 679078677*x^9 + 18844334727*x^10 +...
such that
x/Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3 = 1 + x + 2*x^2 + 14*x^3 + 154*x^4 + 2222*x^5 + 38778*x^6 + 783918*x^7 + 17885082*x^8 + 452719118*x^9 + ...
ITERATIONS OF x*A(x).
Let G(x) = x*A(x), then
A(x) = 1 + G(x)/3 + G(G(x))*2/3^2 + G(G(G(x)))*2^2/3^3 + G(G(G(G(x))))*2^3/3^4 + G(G(G(G(G(x)))))*2^4/3^5 +...
The table of coefficients in the iterations of x*A(x) begin:
[1, 1, 3, 21, 231, 3333, 58167, 1175877, 26827623, ...];
[1, 2, 8, 58, 630, 8958, 154530, 3096330, 70161318, ...];
[1, 3, 15, 117, 1285, 18167, 310735, 6177745, 139076385, ...];
[1, 4, 24, 204, 2308, 32800, 559124, 11053668, 247451528, ...];
[1, 5, 35, 325, 3835, 55365, 946623, 18671961, 416326935, ...];
[1, 6, 48, 486, 6026, 89158, 1539350, 30423134, 677231222, ...];
[1, 7, 63, 693, 9065, 138383, 2427943, 48304893, 1076756889, ...];
[1, 8, 80, 952, 13160, 208272, 3733608, 75127944, 1682704256, ...];
[1, 9, 99, 1269, 18543, 305205, 5614887, 114768093, 2592154167, ...]; ...
in which the following sum along column k equals a(k+1):
a(2) = 3 = 1/3 + 2*2/9 + 3*4/27 + 4*8/81 + 5*16/243 + 6*32/729 +...
a(3) = 21 = 3/3 + 8*2/9 + 15*4/27 + 24*8/81 + 35*16/243 + 48*32/729 + ...
a(4) = 231 = 21/3 + 58*2/9 + 117*4/27 + 204*8/81 + 325*16/243 + 486*32/729 +...
a(5) = 3333 = 231*2/3 + 630*2/9 + 1285*4/27 + 2308*8/81 + 3835*16/243 + 6026*32/729 +...
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