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A165327 E.g.f: Sum_{n>=0} 2^(n(n-1)) * exp(2^n*x) * x^n/n!. 6

%I

%S 1,2,9,125,6561,1419857,1291467969,4902227890625,76686282021340161,

%T 4891005035897482905857,1262332172765951010966606849,

%U 1312086657801266767978668212890625

%N E.g.f: Sum_{n>=0} 2^(n(n-1)) * exp(2^n*x) * x^n/n!.

%C 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.

%F a(n) = (2^(n-1) + 1)^n.

%e E.g.f: A(x) = 1 + 2*x + 3^2*x^2/2! + 5^3*x^3/3! + 9^4*x^4/4! +...

%e A(x) = exp(x) + exp(2x)*x + 2^2*exp(4x)*x^2/2! + 2^6*exp(8x)*x^3/3! +...

%e This is a special case of the more general statement:

%e 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.

%o (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)}

%Y Cf. variants: A136516, A055601, A079491.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 15 2009

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Last modified May 21 14:53 EDT 2019. Contains 323443 sequences. (Running on oeis4.)