login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A192667 E.g.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^3 - 1)^n/n!. 1
1, 1, 6, 99, 2616, 95625, 4468608, 254426571, 17087348736, 1322490908817, 115902895680000, 11345706419279859, 1226971723559141376, 145275861381024623769, 18691551435638516649984, 2596726179631913433046875, 387404350400960574932385792 (list; graph; refs; listen; history; text; internal format)
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: A187522 A224615 A138913 * A127636 A281502 A226413

Adjacent sequences:  A192664 A192665 A192666 * A192668 A192669 A192670

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 13 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 4 12:05 EST 2020. Contains 338923 sequences. (Running on oeis4.)