A206403 E.g.f. A(x) satisfies: exp(A(x)) = 2*exp(A(x)^2) - (1-x), with A(0) = 0.

1, 3, 26, 398, 8604, 239296, 8135504, 326921192, 15159790680, 796766681184, 46805302872624, 3039065898588144, 216125148650657232, 16706734205424667296, 1394789126514873632832, 125073511937467759505760, 11989203887017099078716384, 1223407961244225521367780096

Offset

1,2

Links

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

Formula

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

a(n) ~ sqrt(s/(exp(s)*(1-s+2*s^2))) * n^(n-1) / (exp(n) * (1+exp(s)*(1-1/(2*s)))^(n-1/2)), where s = 0.30957459575023142183... is the root of the equation exp(s) = 4*s*exp(s^2). - Vaclav Kotesovec, Jan 12 2014

Example

E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 398*x^4/4! + 8604*x^5/5! +...
where A( 1 + exp(x) - 2*exp(x^2) ) = x.
Related expansions:
exp(A(x)) = 1 + x + 4*x^2/2! + 36*x^3/3! + 548*x^4/4! + 11800*x^5/5! +...
2*exp(A(x)^2) = 2 + 4*x^2/2! + 36*x^3/3! + 548*x^4/4! + 11800*x^5/5! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[1 + Exp[x] - 2*Exp[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(1+exp(X)-2*exp(X^2)), n))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A138014, A206401, A206402, A206404, A206405.

Sequence in context: A143155 A300283 A326431 * A192554 A306280 A305144

Adjacent sequences: A206400 A206401 A206402 * A206404 A206405 A206406

Keyword

nonn

Author

Paul D. Hanna, Feb 07 2012

Status

approved