A200319 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2).

1, 2, 12, 132, 2040, 40440, 979440, 28034160, 925858080, 34654465440, 1449705660480, 67029745527360, 3394417068282240, 186842736763562880, 11107390768144070400, 709223357051739129600, 48408150749346010022400, 3517279496138031162739200, 271050342684747077612160000

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

E.g.f.: Series_Reversion(1+x - exp(x^2)).

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n / n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n/x / n! ).

a(n) ~ (c/2)^(1/4) * n^(n-1) / (sqrt(1+c) * exp(n) * (1+sqrt(c/2)-1/sqrt(2*c))^(n-1/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

Example

E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
where A(1+x - exp(x^2)) = x and A(x) = x-1 + exp(A(x)^2).

Mathematica

Rest[CoefficientList[InverseSeries[Series[1 - E^x^2 + x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(1+x-exp(x^2+x^2*O(x^n))), n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (exp(x^2+x*O(x^n))-1)^m)/m!); n!*polcoeff(A, n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (exp(x^2+x*O(x^n))-1)^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

Crossrefs

Cf. A200320, A213640, A202356.

Sequence in context: A286422 A073551 A215363 * A213640 A266489 A208830

Adjacent sequences: A200316 A200317 A200318 * A200320 A200321 A200322

Keyword

nonn

Author

Paul D. Hanna, Nov 15 2011

Status

approved