A340913 E.g.f.: Sum_{n>=0} x^n * exp( 2*x*exp(n*x) ) / n!.

1, 3, 9, 39, 249, 2083, 21309, 258891, 3676465, 60188355, 1120986549, 23477257147, 547630675689, 14116139238243, 399459984039853, 12338912081665707, 413926973626402401, 15012208323417943171, 586241726429900170341, 24560642734838666902107

Offset

0,2

Comments

The e.g.f. A(x) of this sequence is motivated by the following identity:

Sum_{n>=0} p^n/n! * exp(q*r^n) = Sum_{n>=0} q^n/n! * exp(p*r^n) ;

here, p = x, q = 2*x, and r = exp(x).

Links

Table of n, a(n) for n=0..19.

Formula

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

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

Example

E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 39*x^3/3! + 249*x^4/4! + 2083*x^5/5! + 21309*x^6/6! + 258891*x^7/7! + 3676465*x^8/8! + 60188355*x^9/9! + ...
where
A(x) = exp(2*x) + x*exp(2*x*exp(x)) + x^2*exp(2*x*exp(2*x))/2! + x^3*exp(2*x*exp(3*x))/3! + x^4*exp(2*x*exp(4*x))/4! + x^5*exp(2*x*exp(5*x))/5! + ...
also
A(x) = exp(x) + 2*x*exp(x*exp(x)) + 2^2*x^2*exp(x*exp(2*x))/2! + 2^3*x^3*exp(x*exp(3*x))/3! + 2^4*x^4*exp(x*exp(4*x))/4! + 2^5*x^5*exp(x*exp(5*x))/5! + ...

Prog

(PARI) {a(n) = my(A=1); A = sum(m=0, n, (x^m/m!)*exp(2*x*exp(m*x +x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = sum(m=0, n, (2^m*x^m/m!)*exp(x*exp(m*x +x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A340909, A340332, A340335.

Sequence in context: A273396 A058105 A090012 * A079096 A143293 A376403

Adjacent sequences: A340910 A340911 A340912 * A340914 A340915 A340916

Keyword

nonn

Author

Paul D. Hanna, Jan 30 2021

Status

approved