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EXAMPLE
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E.g.f.: A(x) = 1 + x + 7*x^2/2! + 187*x^3/3! + 11517*x^4/4! + 1269821*x^5/5! + 218962723*x^6/6! + 54377141463*x^7/7! + 18394396344313*x^8/8! + 8139652855993369*x^9/9! + 4568235711128252991*x^10/10! +...
such that
A(x) = 1 + x/A(x) + 3^2*(x/A(x))^2/2! + 4^4*(x/A(x))^3/3! + 5^6*(x/A(x))^4/4! + 6^8*(x/A(x))^5/5! + 7^10*(x/A(x))^6/6! +...+ (n+1)^(2*n-2) * (x/A(x))^n/n! +...
The table of coefficients of x^k/k! in A(x)^(n+1) begins:
[1, 1, 7, 187, 11517, 1269821, 218962723, 54377141463, ...];
[1, 2, 16, 416, 24824, 2680992, 456281248, 112343845952, ...];
[1, 3, 27, 693, 40173, 4249143, 713494215, 174131360553, ...];
[1, 4, 40, 1024, 57840, 5991584, 992282944, 239988068352, ...];
[1, 5, 55, 1415, 78125, 7927425, 1294484035, 310180891235, ...];
[1, 6, 72, 1872, 101352, 10077696, 1622102688, 384996798528, ...];
[1, 7, 91, 2401, 127869, 12465467, 1977326743, 464744426517, ...];
[1, 8, 112, 3008, 158048, 15115968, 2362541440, 549755813888, ...]; ...
in which the main diagonal begins:
[1, 2, 27, 1024, 78125, 10077696, 1977326743, ..., (n+1)^(2*n-1), ...].
RELATED SERIES.
log(A(x)) = x + 6*x^2/2! + 168*x^3/3! + 10700*x^4/4! + 1203960*x^5/5! + 210264432*x^6/6! + 52655421952*x^7/7! + 17914652980128*x^8/8! + 7960047283278720*x^9/9! + 4481097300680675840*x^10/10! +...
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