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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n) * x^n / 2^(n^2) is defined by
(1) A(x) = Sum_{n>=0} 2^n * x^n * (3^n - 2^n)^n / (2^n - 3^n*x)^(n+1),
(2) A(x) = Sum_{n>=0} 2^n * x^n * (3^n + 2^n)^n / (2^n + 3^n*x)^(n+1),
(3) A(x) = Sum_{n>=0} x^n * ((3/2)^n - 1)^n / (1 - (3/2)^n*x)^(n+1),
(4) A(x) = Sum_{n>=0} x^n * ((3/2)^n + 1)^n / (1 + (3/2)^n*x)^(n+1).
FORMULAS FOR TERMS.
a(n) = Sum_{k=0..n} binomial(n, k) * (3^k - 2^k)^k * (2^n * 3^k)^(n-k).
a(n) = Sum_{k=0..n} binomial(n, k) * (3^k + 2^k)^k * (-2^n * 3^k)^(n-k).
a(n) = 2^(n^2) * Sum_{k=0..n} binomial(n, k) * ((3/2)^n - (3/2)^k)^(n-k).
a(n) = 2^(n^2) * Sum_{k=0..n} binomial(n, k) * ((3/2)^n + (3/2)^k)^(n-k) * (-1)^k.
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EXAMPLE
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G.f. A(x) = Sum_{n>=0} a(n) * x^n / 2^(n^2) begins:
A(x) = 1 + 3/2*x + 65/2^4*x^2 + 14499/2^9*x^3 + 33321281/2^16*x^4 + 707203658403/2^25*x^5 + 133454900550883025/2^36*x^6 + 222607516710114412650819/2^49*x^7 + 3290276040925391895704354602241/2^64*x^8 + 432761522266798203207806930832349560003/2^81*x^9 + ...
such that
A(x) = 1/(1 + x) + 2*x*(3 + 2)/(2 + 3*x)^2 + 2^2*x^2*(3^2 + 2^2)^2/(2^2 + 3^2*x)^3 + 2^3*x^3*(3^3 + 2^3)^3/(2^3 + 3^3*x)^4 + 2^4*x^4*(3^4 + 2^4)^4/(2^4 + 3^4*x)^5 + 2^5*x^5*(3^5 + 2^5)^5/(2^5 + 3^5*x)^6 + 2^6*x^6*(3^6 + 2^6)^6/(2^6 + 3^6*x)^7 + ...
also
A(x) = 1/(1 - x) + 2*x*(3 - 2)/(2 - 3*x)^2 + 2^2*x^2*(3^2 - 2^2)^2/(2^2 - 3^2*x)^3 + 2^3*x^3*(3^3 - 2^3)^3/(2^3 - 3^3*x)^4 + 2^4*x^4*(3^4 - 2^4)^4/(2^4 - 3^4*x)^5 + 2^5*x^5*(3^5 - 2^5)^5/(2^5 - 3^5*x)^6 + 2^6*x^6*(3^6 - 2^6)^6/(2^6 - 3^6*x)^7 + ...
More explicitly,
A(x) = 1/(1 + x) + 2*x*5/(2 + 3*x)^2 + 2^2*x^2*13^2/(2^2 + 3^2*x)^3 + 2^3*x^3*35^3/(2^3 + 3^3*x)^4 + 2^4*x^4*97^4/(2^4 + 3^4*x)^5 + 2^5*x^5*275^5/(2^5 + 3^5*x)^6 + 2^6*x^6*793^6/(2^6 + 3^6*x)^7 + ...
also
A(x) = 1/(1 - x) + 2*x/(2 - 3*x)^2 + 2^2*x^2*5^2/(2^2 - 3^2*x)^3 + 2^3*x^3*19^3/(2^3 - 3^3*x)^4 + 2^4*x^4*65^4/(2^4 - 3^4*x)^5 + 2^5*x^5*211^5/(2^5 - 3^5*x)^6 + 2^6*x^6*665^6/(2^6 - 3^6*x)^7 + ...
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PROG
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(PARI) {a(n) = my(A = sum(m=0, n+1, 2^m * (3^m - 2^m)^m*x^m/(2^m - 3^m*x +x*O(x^n) )^(m+1) )); 2^(n^2)*polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n+1, 2^m * (3^m + 2^m)^m*x^m/(2^m + 3^m*x +x*O(x^n) )^(m+1) )); 2^(n^2)*polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* Binomial formula for terms */
{a(n) = sum(k=0, n, binomial(n, k) * (3^k - 2^k)^k * (2^n * 3^k)^(n-k) )}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* Binomial formula for terms */
{a(n) = sum(k=0, n, binomial(n, k) * (3^k + 2^k)^k * (-2^n * 3^k)^(n-k) )}
for(n=0, 15, print1(a(n), ", "))
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