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EXAMPLE
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E.g.f.: A(x) = 1 + x + 8*x^2/2! + 176*x^3/3! + 6896*x^4/4! + 397888*x^5/5! +...
such that A(x) = exp(8*x*G(x)^7) / G(x)^7
where G(x) = 1 + x*G(x)^8 is the g.f. of A007556:
G(x) = 1 + x + 8*x^2 + 92*x^3 + 1240*x^4 + 18278*x^5 + 285384*x^6 +...
Note that
A'(x) = exp(8*x*G(x)^7) = 1 + 8*x + 176*x^2/2! + 6896*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 7*x^2/2 + 77*x^3/3 + 1015*x^4/4 + 14763*x^5/5 +...
and so A'(x)/A(x) = G(x)^7.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1, 8, 176, 6896, 397888, 30584128, 2948178304, ...];
n=2: [1, 2, 18, 400, 15584, 892896, 68217472, 6543183488, ...];
n=3: [1, 3, 30, 678, 26352, 1501344, 114073632, 10890011520, ...];
n=4: [1, 4, 44, 1016, 39512, 2241472, 169479808, 16107837568, ...];
n=5: [1, 5, 60, 1420, 55400, 3133560, 235931200, 22331561600, ...];
n=6: [1, 6, 78, 1896, 74376, 4200048, 315106128, 29713474944, ...];
n=7: [1, 7, 98, 2450, 96824, 5465656, 408881872, 38425052848, ...];
n=8: [1, 8, 120, 3088, 123152, 6957504, 519351232, 48658878080, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 30, 1016, 55400, 4200048, 408881872, 48658878080, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 8^(n-6) * (n+1)^(n-7) * (16807*n^6 + 143031*n^5 + 525875*n^4 + 1074745*n^3 + 1294846*n^2 + 876856*n + 262144) for n>=0.
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PROG
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(PARI) {a(n) = local(G=1); for(i=1, n, G=1+x*G^8 +x*O(x^n)); n!*polcoeff(exp(8*x*G^7)/G^7, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0|n==1, 1, sum(k=0, n, 8^k * n!/k! * binomial(8*n-k-8, n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))
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