A180602 a(n) = (2^(n+1) - 1)^n.

1, 3, 49, 3375, 923521, 992436543, 4195872914689, 70110209207109375, 4649081944211090042881, 1227102111503512992112190463, 1291749870339606615892191271170049, 5429914198235566686555216227881787109375

Offset: 0

Author

Paul D. Hanna, Sep 11 2010

Keywords

• nonn
• easy

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

Examples

			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! +...
		

Crossrefs

Cf. A086459 (signed, offset 1), variants: A055601, A079491, A136516, A165327.

Cf. A001694.

Programs

Magma

[(2^(n+1)-1)^n : n in [0..11]]; // Arkadiusz Wesolowski, Aug 26 2013

Maple

A180602:=n->(2^(n+1) - 1)^n: seq(A180602(n), n=0..10); # Wesley Ivan Hurt, Oct 09 2014

Mathematica

Table[(2^(n + 1) - 1)^n, {n, 0, 10}] (* Wesley Ivan Hurt, Oct 09 2014 *)

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)}

Python

def A180602(n): return ((1<Chai Wah Wu, Sep 13 2024

Formula

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

Extensions

Name changed by Arkadiusz Wesolowski, Aug 26 2013