A340332 E.g.f.: Sum_{n>=0} x^n * exp(3*2^n*x) / n!.

1, 4, 22, 172, 1906, 30004, 670042, 21232012, 953930146, 60764655844, 5485191552682, 701608383497212, 127123579686312946, 32624630107479118804, 11856598040266426776442, 6101496890841018365899372, 4445361041794934079330768706, 4585062274218901537813324793284

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 = 3*x, and r = 2.

Links

Table of n, a(n) for n=0..17.

Formula

The e.g.f. satisfies the following relations.

(1) Sum_{n>=0} x^n * exp(3*2^n*x) / n!.

(2) Sum_{n>=0} 3^n*x^n * exp(2^n*x) / n!.

Example

E.g.f.: A(x) = 1 + 4*x + 22*x^2/2! + 172*x^3/3! + 1906*x^4/4! + 30004*x^5/5! + 670042*x^6/6! + 21232012*x^7/7! + 953930146*x^8/8! + 60764655844*x^9/9! + ...
where
A(x) = exp(3*x) + x*exp(3*2*x) + x^2*exp(3*2^2*x)/2! + x^3*exp(3*2^3*x)/3! + x^4*exp(3*2^4*x)/4! + x^5*exp(3*2^5*x)/5! + ...
also
A(x) = exp(x) + 3*x*exp(2*x) + 3^2*x^2*exp(2^2*x)/2! + 3^3*x^3*exp(2^3*x)/3! + 3^4*x^4*exp(2^4*x)/4! + 3^5*x^5*exp(2^5*x)/5! + ...

Prog

(PARI) {a(n) = my(A = sum(m=0, n, x^m/m! * exp(3*2^m*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n, 3^m*x^m/m! * exp(2^m*x +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A340450.

Sequence in context: A350268 A353186 A346665 * A207654 A197923 A294343

Adjacent sequences: A340329 A340330 A340331 * A340333 A340334 A340335

Keyword

nonn

Author

Paul D. Hanna, Jan 09 2021

Status

approved