A192667 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^3 - 1)^n/n!.

1, 1, 6, 99, 2616, 95625, 4468608, 254426571, 17087348736, 1322490908817, 115902895680000, 11345706419279859, 1226971723559141376, 145275861381024623769, 18691551435638516649984, 2596726179631913433046875, 387404350400960574932385792

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

E.g.f. A(x) equals the formal inverse of function (x-1)/exp(x^3-1).

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

E.g.f.: A(x) = 1 + Series_Reversion( x/exp((1+x)^3 - 1) ).

E.g.f. satisfies: A(x/G(x)) = 1 + x where G(x) = exp((1+x)^3 - 1) and G(x) = x/Series_Reversion(A(x)-1) = e.g.f. of A192989.

a(n) ~ n^(n-1) / (3*sqrt(s*(1+s^2)) * exp(n) * r^n), where s = 1/9*(3 + (297/2 - (81*sqrt(13))/2)^(1/3) + 3*((1/2)*(11 + 3*sqrt(13)))^(1/3)) = 1.2228950301592487561... is the root of the equation 3*(s-1)*s^2 = 1, and r = (s-1)*exp(1-s^3) = 0.097309376917122928890... - Vaclav Kotesovec, Feb 26 2014

Example

E.g.f.: A(x) = 1 + x + 6*x^2/2! + 99*x^3/3! + 2616*x^4/4! +...
where (A(x) - 1)/exp(A(x)^3-1) = x
and A(x/G(x)) = 1 + x where G(x) = exp(3*x + 3*x^2 + x^3):
G(x) = 1 + 3*x + 15*x^2/2! + 87*x^3/3! + 585*x^4/4! + 4383*x^5/5! +...
Related expansions.
(A(x)^3-1) = 3*x + 24*x^2/2! + 411*x^3/3! + 11088*x^4/4!  + 410175*x^5/5! +...
(A(x)^3-1)^2 = 18*x^2/2! + 432*x^3/3! + 13320*x^4/4! + 529920*x^5/5! +...
(A(x)^3-1)^3 = 162*x^3/3! + 7776*x^4/4! + 377460*x^5/5! +...
(A(x)^3-1)^4 = 1944*x^4/4! + 155520*x^5/5! +...

Mathematica

CoefficientList[1 + InverseSeries[Series[x/E^((1+x)^3 - 1), {x, 0, 20}], x], x]*Range[0, 20]! (* Vaclav Kotesovec, Feb 26 2014 *)

Prog

(PARI) {a(n)=local(A=1+serreverse(x/exp(3*x+3*x^2+x^3+x^2*O(x^n)))); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*exp(A^3-1+x*O(x^n))); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, (A^3-1+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}

Crossrefs

Cf. A192949, A192989.

Sequence in context: A224615 A138913 A360336 * A360339 A127636 A281502

Adjacent sequences: A192664 A192665 A192666 * A192668 A192669 A192670

Keyword

nonn

Author

Paul D. Hanna, Jul 13 2011

Status

approved