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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 19*x^4 + 104*x^5 + 717*x^6 + 5802*x^7 +...
where, by definition,
A(x) = 1 + x/(A(x) + x) + 2^2*x^2/(A(x) + 2*x)^2 + 3^3*x^3/(A(x) + 3*x)^3 + 4^4*x^4/(A(x) + 4*x)^4 +...+ n^n*x^n/(A(x) + n*x)^n +...
also, g.f. A(x) satisfies:
A(x) = 1 + x/A(x) + 3*x^2/A(x)^2 + 12*x^3/A(x)^3 + 60*x^4/A(x)^4 + 360*x^5/A(x)^5 + 2520*x^6/A(x)^6 +...+ (n+1)!/2*x^n/A(x)^n +...
RELATED TABLES.
Form an array of coefficients of x^k in A(x)^n, which begins:
n=1: [1, 1, 2, 5, 19, 104, 717, 5802, 53337, ...];
n=2: [1, 2, 5, 14, 52, 266, 1743, 13644, 122547, ...];
n=3: [1, 3, 9, 28, 105, 513, 3203, 24201, 211977, ...];
n=4: [1, 4, 14, 48, 185, 880, 5266, 38376, 327252, ...];
n=5: [1, 5, 20, 75, 300, 1411, 8155, 57365, 475650, ...];
n=6: [1, 6, 27, 110, 459, 2160, 12158, 82734, 666567, ...];
n=7: [1, 7, 35, 154, 672, 3192, 17640, 116509, 912086, ...];
n=8: [1, 8, 44, 208, 950, 4584, 25056, 161280, 1227665, ...];
n=9: [1, 9, 54, 273, 1305, 6426, 34965, 220320, 1632960, ...]; ...
then the main diagonal equals n*n!/2 for n > 1:
[1, 2, 9, 48, 300, 2160, 17640, 161280, 1632960, ...].
Form an array of coefficients of x^k in (A(x) + n*x)^n, which begins:
n=1: [1, 2, 2, 5, 19, 104, 717, ...];
n=2: [1, 6, 13, 22, 72, 342, 2159, ...];
n=3: [1, 12, 54, 127, 285, 1116, 6110, ...];
n=4: [1, 20, 158, 640, 1625, 4416, 19746, ...];
n=5: [1, 30, 370, 2425, 9375, 25536, 80155, ...];
n=6: [1, 42, 747, 7310, 43119, 163296, 474326, ...];
n=7: [1, 56, 1358, 18627, 158697, 875980, 3294172, ...]; ...
then the main diagonal equals n^n*(n+1)/2 for n >= 1:
[1, 6, 54, 640, 9375, 163296, 3294172, 75497472, ...].
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