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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +..
Related expansions:
. A(x)/x = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +..
. A(x)^2/x = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +..
. A'(x) = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! +...
. A(A(x)) = x + 4*x^2/2! + 48*x^3/3! + 1008*x^4/4! + 30880*x^5/5! +...
. A(A(x))^2 = 2*x^2/2! + 24*x^3/3! + 480*x^4/4! + 13920*x^5/5! +...
Illustrate a main property of the iterations A_n(x) of A(x) by:
. [A_3(x)]^2 = A(x)^2 * A_2'(x);
. [A_4(x)]^2 = A(x)^2 * A_3'(x);
. [A_5(x)]^2 = A(x)^2 * A_4'(x); ...
which can be shown to hold by the chain rule of differentiation.
...
The RIORDAN ARRAY (A(x)/x, A(x)) begins:
. 1;
. 1, 1;
. 6/2!, 2, 1;
. 78/3!, 14/2!, 3, 1;
. 1648/4!, 192/3!, 24/2!, 4, 1;
. 49500/5!, 4136/4!, 348/3!, 36/2!, 5, 1;
. 1957968/6!, 124840/5!, 7680/4!, 552/3!, 50/2!, 6, 1;
. 97097336/7!, 4928256/6!, 233940/5!, 12520/4!, 810/3!, 66/2!, 7, 1; ...
where the e.g.f. of column k = [A(x)/x]^(k+1) for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
. 0;
. 1, 0;
. 4/2!, 2, 0;
. 42/3!, 8/2!, 3, 0;
. 768/4!, 84/3!, 12/2!, 4, 0;
. 20680/5!, 1536/4!, 126/3!, 16/2!, 5, 0;
. 749040/6!, 41360/5!, 2304/4!, 168/3!, 20/2!, 6, 0;
. 34497792/7!, 1498080/6!, 62040/5!, 3072/4!, 210/3!, 24/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*A(x)^2/x for k>=0.
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