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 A165327 E.g.f: Sum_{n>=0} 2^(n(n-1)) * exp(2^n*x) * x^n/n!. 6
 1, 2, 9, 125, 6561, 1419857, 1291467969, 4902227890625, 76686282021340161, 4891005035897482905857, 1262332172765951010966606849, 1312086657801266767978668212890625 (list; graph; refs; listen; history; text; internal format)
 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. LINKS 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 Cf. variants: A136516, A055601, A079491. Sequence in context: A062457 A307539 A067966 * A090242 A047684 A162955 Adjacent sequences:  A165324 A165325 A165326 * A165328 A165329 A165330 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 15 2009 STATUS approved

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Last modified June 2 14:10 EDT 2020. Contains 334781 sequences. (Running on oeis4.)