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FORMULA
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a(n) = exp(-3^n)*Sum_{k>=0} (3^k*k)^n/k!.
a(n) = [x^n/n! ] Bell(x)^(3^n) where Bell(x) = exp(exp(x) - 1) is the e.g.f. of the Bell numbers.
a(n) = Sum_{k=0..n} S2(n,k)*3^(n*k), where S2(n,k) = A008277(n,k) are the Stirling numbers of the second kind.
G.f.: A(x) = Sum_{n>=0} 3^(n^2)*x^n/[Product_{k=1..n} (1-k*3^n*x)].
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PROG
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(PARI) {a(n)=local(infnty=n^5+10); round(exp(-3^n)*sum(k=0, infnty, (3^k*k)^n/k!))}
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (exp(3^k*x +x*O(x^n))-1)^k/k!), n)}
(PARI) {a(n)=n!*polcoeff(exp(3^n*(exp(x +x*O(x^n))-1)), n)}
(PARI) {S2(n, k)=(1/k!)*sum(i=0, k, (-1)^(k-i)*binomial(k, i)*i^n)}
{a(n)=sum(k=0, n, S2(n, k)*3^(n*k))} (End)
(PARI) {a(n)=polcoeff(sum(k=0, n, (3^k*x)^k/prod(j=1, k, 1-j*3^k*x+x*O(x^n))), n)}
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