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COMMENTS
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In general, we have the o.g.f. identity:
Sum_{n>=0} m^n * q^(n^2) * x^n/(1 - b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n ; here, q=2, m=2, b=1.
In general, we have the e.g.f. identity:
Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n / n! = Sum_{n>=0} (m*q^n + b)^n * x^n / n! ; here, q=2, m=2, b=1.
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PROG
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(PARI) {a(n, q, m, b) = (m*q^n + b)^n}
for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", "))
(PARI) /* E.g.f. formula: */
{a(n, q, m, b) = polcoeff( sum(k=0, n, m^k * q^(k^2) * x^k / (1 - b*q^k*x +x*O(x^n))^(k+1)), n)}
for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", "))
(PARI) /* E.g.f. formula: */
{a(n, q, m, b) = n! * polcoeff( sum(k=0, n, m^k * q^(k^2) * exp(b*q^k*x +x*O(x^n)) * x^k/k!), n)}
for(n=0, 15, print1(a(n, q=2, m=2, b=1), ", "))
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