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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 8*x^4/4! - 95*x^5/5! - 144*x^6/6! +...
where
A(x) = 1 + x/A(x) + 4*(x/A(x))^2/2! + 27*(x/A(x))^3/3! + 256*(x/A(x))^4/4! + 3125*(x/A(x))^5/5! +...+ n^n*(x/A(x))^n/n! +...
If we form a table of coefficients of x^k/k! in A(x)^n, like so:
[1, 1, 2, 3, -8, -95, -144, 5299, 51584, ...];
[1, 2, 6, 18, 32, -150, -1728, -1078, 144384, ...];
[1, 3, 12, 51, 192, 375, -2592, -29841, 12288, ...];
[1, 4, 20, 108, 568, 2500, 5184, -48020, -557056, ...];
[1, 5, 30, 195, 1280, 7845, 38880, 84035, -983040, ...];
[1, 6, 42, 318, 2472, 18750, 129456, 705894, 1572864, ...];
[1, 7, 56, 483, 4312, 38395, 326592, 2485567, 14680064, ...];
[1, 8, 72, 696, 6992, 70920, 704448, 6588344, 54442368, ...];
[1, 9, 90, 963, 10728, 121545, 1368144, 14890995, 150994944, ...]; ...
then the main diagonal equals (k+1)*k^k for k>=0.
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