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FORMULA
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G.f.: Sum_{n>=0} (2^n + i)^n * x^n / (1 + 2^n*i*x)^(n+1).
G.f.: Sum_{n>=0} (2^n - i)^n * x^n / (1 - 2^n*i*x)^(n+1).
a(n) = Sum_{k=0..n} i^k * binomial(n,k) * (2^n - 2^k*i)^(n-k).
a(n) = Sum_{k=0..n} (-i)^k * binomial(n,k) * (2^n + 2^k*i)^(n-k).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 18*x^2 + 560*x^3 + 68642*x^4 + 34215232*x^5 + 69224885118*x^6 + 564395560679680*x^7 + 18462516899827251202*x^8 + ...
such that
A(x) = 1/(1+i*x) + (2 + i)*x/(1 + 2*i*x)^2 + (2^2 + i)^2*x^2/(1 + 2^2*i*x)^3 + (2^3 + i)^3*x^3/(1 + 2^3*i*x)^4 + (2^4 + i)^4*x^4/(1 + 2^4*i*x)^5 + (2^5 + i)^5*x^5/(1 + 2^5*i*x)^6 + (2^6 + i)^6*x^6/(1 + 2^6*i*x)^7 + ...
Also,
A(x) = 1/(1-i*x) + (2 - i)*x/(1 - 2*i*x)^2 + (2^2 - i)^2*x^2/(1 - 2^2*i*x)^3 + (2^3 - i)^3*x^3/(1 - 2^3*i*x)^4 + (2^4 - i)^4*x^4/(1 - 2^4*i*x)^5 + (2^5 - i)^5*x^5/(1 - 2^5*i*x)^6 + (2^6 - i)^6*x^6/(1 - 2^6*i*x)^7 + ...
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PROG
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(PARI) {a(n) = my(A = sum(m=0, n+1, (2^m + I)^m*x^m/(1 + I*2^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, (2^m - I)^m*x^m/(1 - I*2^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, I^k * binomial(n, k) * (2^n - 2^k*I)^(n-k) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, (-I)^k * binomial(n, k) * (2^n + 2^k*I)^(n-k) )}
for(n=0, 20, print1(a(n), ", "))
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