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MAPLE
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with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(9), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(9, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(9, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
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