A206402 E.g.f. A(x) satisfies: exp(A(x)) = x + exp(2*A(x)^2), with A(0) = 0.

1, 3, 26, 422, 9684, 284536, 10205264, 432507008, 21149344320, 1172055816864, 72593488746624, 4969455399927168, 372585629959484928, 30363657581135890176, 2672420848359072517632, 252632488649577779398656, 25529176319428221234402816, 2746226455049097879478060032

Offset

1,2

Links

Table of n, a(n) for n=1..18.

Formula

E.g.f.: A(x) = Series_Reversion( exp(x) - exp(2*x^2) ).

a(n) ~ 1/2 * sqrt((4*s - 1)/(1 - s + 4*s^2)) * n^(n-1) / (exp(s+1)-exp(2*s^2+1))^n, where s = 0.28268257266202691... is the root of the equation exp(s) = 4*exp(2*s^2)*s. - Vaclav Kotesovec, Jan 12 2014

Example

E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 422*x^4/4! + 9684*x^5/5! +...
where A( exp(x) - exp(2*x^2) ) = x.
Related expansions:
exp(A(x)) = 1 + x + 4*x^2/2! + 36*x^3/3! + 572*x^4/4! + 13000*x^5/5! +...
exp(2*A(x)^2) = 1 + 4*x^2/2! + 36*x^3/3! + 572*x^4/4! + 13000*x^5/5! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[E^x - E^(2*x^2), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n)); if(n<1, 0, n!*polcoeff(serreverse(exp(X)-exp(2*X^2)), n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A138014, A206401, A206403, A206404, A206405.

Sequence in context: A192554 A306280 A305144 * A300954 A122949 A305113

Adjacent sequences: A206399 A206400 A206401 * A206403 A206404 A206405

Keyword

nonn

Author

Paul D. Hanna, Feb 07 2012

Status

approved