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EXAMPLE
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G.f.: A(x) = 1 + x + 7*x^2 + 17*x^3 + 55*x^4 + 113*x^5 + 135*x^6 + 321*x^7 + 2103*x^8 + 3217*x^9 + 8295*x^10 + 18145*x^11 + 33687*x^12 + ...
The table of coefficients of x^k in A(2*x)^n begins:
n=1: [1, 2, 28, 136, 880, 3616, 8640, ...];
n=2: [1, 4, 60, 384, 3088, 18368, 99520, ...];
n=3: [1, 6, 96, 752, 6960, 50592, 350848, ...];
n=4: [1, 8, 136, 1248, 12848, 107520, 864000, ...];
n=5: [1, 10, 180, 1880, 21120, 197312, 1765760, ...];
n=6: [1, 12, 228, 2656, 32160, 329088, 3210624, ...];
n=7: [1, 14, 280, 3584, 46368, 512960, 5383168, ...];
n=8: [1, 16, 336, 4672, 64160, 760064, 8500480, ...];
...
from which each term in row n may be reduced modulo 2^n to form the following triangle (trailing zeros suppressed):
A(2*x) (mod 2): [1];
A(2*x)^2 (mod 2^2): [1, 0];
A(2*x)^3 (mod 2^3): [1, 6, 0];
A(2*x)^4 (mod 2^4): [1, 8, 8, 0];
A(2*x)^5 (mod 2^5): [1, 10, 20, 24, 0];
A(2*x)^6 (mod 2^6): [1, 12, 36, 32, 32, 0];
A(2*x)^7 (mod 2^7): [1, 14, 24, 0, 32, 64, 0];
A(2*x)^8 (mod 2^8): [1, 16, 80, 64, 160, 0, 0, 0];
A(2*x)^9 (mod 2^9): [1, 18, 396, 296, 464, 224, 320, 384, 0];
A(2*x)^10 (mod 2^10): [1, 20, 460, 192, 624, 832, 64, 512, 512, 0];
A(2*x)^11 (mod 2^11): [1, 22, 528, 784, 80, 1120, 1664, 1536, 1536, 1024, 0];
A(2*x)^12 (mod 2^12): [1, 24, 600, 2592, 3920, 3328, 2048, 2048, 3584, 0, 0, 0]; ...
The row sums of the above triangle form this sequence.
SPECIFIC VALUES.
A(1/3) = 5.16342352287698185339110618005914219821...
A(1/4) = 2.39174482766273066991171836527389776948...
A(1/5) = 1.76139991234861303660669400163149230805...
A(1/6) = 1.50260759835187695870739455561470784874...
A(t) = 2 at t = 0.22255799946572991869110865358904969874...
A(t) = 3 at t = 0.27939565401912059198552904564135101228...
A(t) = 4 at t = 0.31035599522228672966119371322774606931...
A(t) = 5 at t = 0.33063501311279486918626289986794151717...
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