OFFSET
0,2
COMMENTS
More generally, 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! for all q, m, b.
FORMULA
a(n) = (2^(n-1) + 1)^n.
EXAMPLE
E.g.f: A(x) = 1 + 2*x + 3^2*x^2/2! + 5^3*x^3/3! + 9^4*x^4/4! +...
A(x) = exp(x) + exp(2x)*x + 2^2*exp(4x)*x^2/2! + 2^6*exp(8x)*x^3/3! +...
This is a special case of the more general statement:
Sum_{n>=0} m^n * F(q^n*x)^b * log(F(q^n*x) )^n / n! = Sum_{n>=0} x^n * [y^n] F(y)^(m*q^n + b) where F(x) = exp(x), q=2, m=1/2, b=1.
PROG
(PARI) {a(n, q=2, m=1/2, b=1)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 15 2009
STATUS
approved