OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
E.g.f. also satisfies:
(1) A(x) = x*exp(2*x - A(x)) / cosh(2*x - A(x)).
(2) A(x) = (2*x - A(x)) * exp(4*x - 2*A(x)).
(3) A(x) = Series_Reversion( (2*x + LambertW(2*x))/4 ).
(4) A(x) = Series_Reversion( x - Sum_{n>=2} 2^(n-2) * n^(n-1) * (-x)^n/n! ).
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! - 6*x^3/3! + 16*x^4/4! + 40*x^5/5! - 1584*x^6/6! + 22848*x^7/7! - 225280*x^8/8! + 600192*x^9/9! + 44396800*x^10/10! - 1523498240*x^11/11! + 31443941376*x^12/12! +...
such that A( x*exp(x)*cosh(x) ) = x*exp(2*x).
RELATED SERIES.
Series_Reversion( A(x) ) = x - 2*x^2 + 18*x^3 - 256*x^4 + 5000*x^5 - 124416*x^6 + 3764768*x^7 - 134217728*x^8 +...+ -2^(n-2) * n^(n-1) * (-x)^n/n! +...
which equals (2*x + LambertW(2*x))/4.
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[(2*x + LambertW[2*x])/4, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Sep 11 2016 *)
PROG
(PARI) /* From A( x*exp(x)*cosh(x) ) = x*exp(2*x) */
{a(n) = my(A, X = x + x*O(x^n)); A = subst( x*exp(2*X), x, serreverse( x*exp(X)*cosh(X) )); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* From Series_Reversion( (2*x + LambertW(2*x))/4 ) */
{a(n) = my(A, LambertW_2x = subst( serreverse(x*exp(x + x*O(x^n))), x, 2*x) ); A = serreverse( (2*x + LambertW_2x)/4 ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Sep 05 2016
STATUS
approved