OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..350
FORMULA
E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) + A(x)^3 dx ) with A(0)=1.
(2) A(x) = (1 + B(x))/(1 - B(x)) where B(x) = tan(1-x - 1/A(x)).
(3) log(A(x)) = Series_Reversion( 1-exp(-x) - atan(tanh(x/2)) ).
(4) A( 1-exp(-x) - atan(tanh(x/2)) ) = exp(x).
a(n) ~ n! / (GAMMA(1/3) * 3^(1/3) * n^(2/3) * (1-Pi/4)^(n+1/3)). - Vaclav Kotesovec, Jan 26 2014
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 1968*x^4/4! + 39488*x^5/5! +...
Related expansions.
A(x)^2 = 1 + 4*x + 32*x^2/2! + 400*x^3/3! + 6848*x^4/4! + 149056*x^5/5! +...
A(x)^4 = 1 + 8*x + 96*x^2/2! + 1568*x^3/3! + 32640*x^4/4! + 828032*x^5/5! +...
The logarithm of e.g.f. A(x) begins:
log(A(x)) = 2*x + 8*x^2/2! + 72*x^3/3! + 992*x^4/4! + 18336*x^5/5! +...
and equals Integral A(x) + A(x)^3 dx, where
A(x)^3 = 1 + 6*x + 60*x^2/2! + 864*x^3/3! + 16368*x^4/4! + 385344*x^5/5! +...
MATHEMATICA
CoefficientList[Exp[InverseSeries[Series[1-Exp[-x]-ArcTan[Tanh[x/2]], {x, 0, 20}], x]], x]*Range[0, 20]! (* Vaclav Kotesovec, Dec 20 2013 *)
PROG
(PARI) /* Explicit formula: */
{a(n)=local(A, X=x+x^2*O(x^n)); A=exp(serreverse(1-exp(-X) - atan(tanh(X/2)))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* By definition: A'(x) = A(x)^2 + A(x)^4: */
{a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A^2+A^4+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From: A(x) = exp( Integral A(x) + A(x)^3 dx ): */
{a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(A+A^3)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 18 2013
STATUS
approved