OFFSET
0,2
COMMENTS
More generally, we have the following identities:
(1) 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);
(2) 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.
This sequence results from (2) when q=2, m=2, b=-1.
For n >= 2, a(n) is the first number in a set of three powerful numbers in arithmetic progression with difference a(n)*(2^n - 1). - Arkadiusz Wesolowski, Aug 26 2013
FORMULA
E.g.f.: Sum_{n>=0} 2^(n^2+n) * exp(-2^n*x) * x^n/n!.
EXAMPLE
E.g.f: A(x) = 1 + 3*x + 7^2*x^2/2! + 15^3*x^3/3! + 31^4*x^4/4! +...
A(x) = exp(-x) + 2^2*exp(-2*x)*x + 2^6*exp(-4*x)*x^2/2! + 2^12*exp(-8*x)*x^3/3! +...
MAPLE
MATHEMATICA
Table[(2^(n + 1) - 1)^n, {n, 0, 10}] (* Wesley Ivan Hurt, Oct 09 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, 2^(k^2+k)*exp(-2^k*x+x*O(x^n))*x^k/k!), n)}
(Magma) [(2^(n+1)-1)^n : n in [0..11]]; // Arkadiusz Wesolowski, Aug 26 2013
(Python)
def A180602(n): return ((1<<n+1)-1)**n # Chai Wah Wu, Sep 13 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Sep 11 2010
EXTENSIONS
Name changed by Arkadiusz Wesolowski, Aug 26 2013
STATUS
approved