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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 10*x^2 + 316*x^3 + 49286*x^4 + 29159004*x^5 + 64306390660*x^6 + 545236870010872*x^7 + 18158564638452610374*x^8 + 2398983772627848027521708*x^9 + 1262702849939184484481521481260*x^10 +...
such that the coefficient of x^n in A(x)^(2^n) equals 2^(n*(n+1)) for n>=0.
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^n in A(x)^(2^n) begins:
n=0: [1, 2, 10, 316, 49286, 29159004, 64306390660, ...];
n=1: [1, 4, 24, 672, 99936, 58521472, 128730502912, ...];
n=2: [1, 8, 64, 1536, 205824, 117874688, 257934426112, ...];
n=3: [1, 16, 192, 4096, 440320, 239239168, 517783552000, ...];
n=4: [1, 32, 640, 14336, 1048576, 494141440, 1043408617472, ...];
n=5: [1, 64, 2304, 69632, 3424256, 1073741824, 2119989985280, ...];
n=6: [1, 128, 8704, 434176, 21069824, 2906652672, 4398046511104, ...];
n=7: [1, 256, 33792, 3096576, 229048320, 18765316096, 10095488401408, 72057594037927936, ...]; ...
in which the main diagonal equals 2^(n^2+n):
[1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, ..., 4^(n*(n+1)/2), ...].
RELATED SERIES.
log(A(x)) = 2*x + 16*x^2/2 + 896*x^3/3 + 194560*x^4/4 + 145293312*x^5/5 + 385486422016*x^6/6 + 3815756479332352*x^7/7 + 145259790155527487488*x^8/8 + 21590527069867423236620288*x^9/9 + 12626980518625294860075743051776*x^10/10 +...
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