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EXAMPLE
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E.g.f. A(x) = 1 + 2*x - 26*x^2/2! + 366*x^3/3! - 6270*x^4/4! + 99922*x^5/5! - 1630730*x^6/6! - 33526706*x^7/7! + 1685562866*x^8/8! - 390576999182*x^9/9! + ...
where
1 + 4*x = exp(-1) * (1 + A(x) + A(x)^4/2! + A(x)^9/3! + A(x)^16/4! + A(x)^25/5! + A(x)^36/6! + A(x)^49/7! + ... + A(x)^(n^2)/n! + ...).
RELATED TABLE.
The table of coefficients of x^k/k! in A(x)^(n^2) begins:
n=0: [1, 0, 0, 0, 0, 0, 0, ...];
n=1: [1, 2, -26, 366, -6270, 99922, -1630730, ...];
n=2: [1, 8, -56, -216, 19800, -706472, 14847688, ...];
n=3: [1, 18, 54, -3906, 34290, 1326978, -99273402, ...];
n=4: [1, 32, 544, -4704, -308640, 6962272, 154469920, ...];
n=5: [1, 50, 1750, 25950, -936750, -37790750, 1459186150, ...];
n=6: [1, 72, 4104, 159336, 1906200, -192221928, -7838021880, ...];
n=7: [1, 98, 8134, 535374, 23730210, 239390578, -52296366122, ...]; ...
in which infinite sums of terms along the columns may be illustrated by:
1 = (1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + ...)/e;
4 = (0 + 2 + 8/2! + 18/3! + 32/4! + 50/5! + ...)/e;
0 = (0 + -26 + -56/2! + 54/3! + 544/4! + 1750/5! + ...);
0 = (0 + 366 + -216/2! + -3906/3! + -4704/4! + 25950/5! + ...);
0 = (0 + -6270 + 19800/2! + 34290/3! + -308640/4! + -936750/5! + ...);
0 = (0 + 99922 + -706472/2! + 1326978/3! + 6962272/4! + -37790750/5! ...); ...
and can be used to determine all the terms of this sequence.
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