A274393 E.g.f. A(x) satisfies: A( A( x^3*exp(-3*x) )^(1/3) ) = x.

1, 2, 9, 56, 545, 6696, 100009, 1756112, 35480673, 811332080, 20696592521, 583009540488, 17972297981521, 601961695890296, 21765379980020265, 844991974575946016, 35056808550027808961, 1547900555791042958688, 72474037424646843142153, 3586609339433026549298840, 187062738581835989450074161, 10255505482370456224398408872, 589611389520200188085133153449

Offset

1,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..175 (terms 1..100 from Paul D. Hanna)

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:

(1) A( A( x^3*exp(3*x) )^(1/3) ) = -LambertW(-x*exp(x)).

(2) A(x) = Series_Reversion( A( x^3*exp(-3*x) )^(1/3) ).

(3) A( A(x)^3 * exp(-3*A(x)) ) = x^3.

a(n) ~ c * n! * d^n / n^(3/2), where d = 2.67119188675..., c = 0.3883596303... . - Vaclav Kotesovec, Aug 12 2021

Example

E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 56*x^4/4! + 545*x^5/5! + 6696*x^6/6! + 100009*x^7/7! + 1756112*x^8/8! + 35480673*x^9/9! + 811332080*x^10/10! + 20696592521*x^11/11! + 583009540488*x^12/12! + 17972297981521*x^13/13! + 601961695890296*x^14/14! + 21765379980020265*x^15/15! +...
such that A( A( x^3*exp(-3*x) )^(1/3) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
A( x^3*exp(-3*x) )^(1/3) = x - 2*x^2/2! + 3*x^3/3! + 4*x^4/4! - 155*x^5/5! + 1914*x^6/6! - 15953*x^7/7! + 33592*x^8/8! + 2425257*x^9/9! - 71955530*x^10/10! + 1307665051*x^11/11! - 13707439692*x^12/12! - 125013414227*x^13/13! + 11742108426034*x^14/14! - 370418656051065*x^15/15! +...
Compare the above series reversion to the following series:
A(x)^3 * exp(-3*A(x)) = x^3 - 2*x^6/2! + 3*x^9/3! + 4*x^12/4! - 155*x^15/5! + 1914*x^18/6! - 15953*x^21/7! + 33592*x^24/8! +...
where A( A(x)^3 * exp(-3*A(x)) ) = x^3.
Note the following series is also in powers of x^3:
A(-A(x)^3 * exp(-3*A(x)) ) = -x^3 + 4*x^6/2! - 24*x^9/3! + 208*x^12/4! - 2400*x^15/5! + 36432*x^18/6! - 700672*x^21/7! + 16221088*x^24/8! - 434076480*x^27/9! + 13091390560*x^30/10! - 438602465664*x^33/11! + 16177344184080*x^36/12! - 652301794869088*x^39/13! + 28571154198355888*x^42/14! +...

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^3*exp(-3*x +x*O(x^n)))^(1/3) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A274275, A274394, A274395.

Sequence in context: A277541 A033917 A277482 * A371196 A004103 A295775

Adjacent sequences: A274390 A274391 A274392 * A274394 A274395 A274396

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2016

Status

approved