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FORMULA
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a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (sum {d|i} (d*s_d))^(s_i*(i*s_i+1)/2) * (-1 + sum {d|i} (d*s_d))^(s_i*(i*s_i-1)/2) or {i=j, even} (sum {d|i and d/i is odd} (d*s_d))^s_i * (sum {d|i} (d*s_d))^(i*s_i^2/2) * (-1 + sum {d|i} (d*s_d))^(s_i*(i*s_i-2)/2) or {i < j} (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j) or {i > j} (-1 + sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)
a(n) is asymptotic to (n^binomial(n+1, 2) * (n-1)^binomial(n, 2))/n! = A079189(n)/A000142(n)
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