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FORMULA
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G.f.: Sum_{n>=0} (3^n + 1)^n * x^n / (1 + 3^n*x)^(n+1).
G.f.: Sum_{n>=0} (3^n - 1)^n * x^n / (1 - 3^n*x)^(n+1).
a(n) = Sum_{k=0..n} binomial(n,k) * (3^n - 3^k)^(n-k).
a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * (3^n + 3^k)^(n-k).
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 77*x^2 + 19359*x^3 + 42889529*x^4 + 846714983643*x^5 + 150077691692889797*x^6 + 239295126680408718226359*x^7 + ...
such that
A(x) = 1/(1+x) + (3 + 1)*x/(1 + 3*x)^2 + (3^2 + 1)^2*x^2/(1 + 3^2*x)^3 + (3^3 + 1)^3*x^3/(1 + 3^3*x)^4 + (3^4 + 1)^4*x^4/(1 + 3^4*x)^5 + (3^5 + 1)^5*x^5/(1 + 3^5*x)^6 + (3^6 + 1)^6*x^6/(1 + 3^6*x)^7 + ...
Also,
A(x) = 1/(1-x) + (3 - 1)*x/(1 - 3*x)^2 + (3^2 - 1)^2*x^2/(1 - 3^2*x)^3 + (3^3 - 1)^3*x^3/(1 - 3^3*x)^4 + (3^4 - 1)^4*x^4/(1 - 3^4*x)^5 + (3^5 - 1)^5*x^5/(1 - 3^5*x)^6 + (3^6 - 1)^6*x^6/(1 - 3^6*x)^7 + ...
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PROG
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(PARI) {a(n) = my(A = sum(m=0, n+1, (3^m + 1)^m*x^m/(1 + 3^m*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n+1, (3^m - 1)^m*x^m/(1 - 3^m*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (3^n - 3^k)^(n-k) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, (-1)^k * binomial(n, k) * (3^n + 3^k)^(n-k) )}
for(n=0, 20, print1(a(n), ", "))
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