OFFSET
1,3
COMMENTS
Compare to: G(x) = sin(x + Integral G(x) dx) holds when G(x) = x*(4+2*x-x^3)/(4+x^4).
FORMULA
E.g.f. A(x) satisfies:
(1) A'(x) = (1 + A(x))*sqrt(1 + A(x)^2).
(2) A(x) = G(x)*(2 - G(x))/(2 - 2*G(x)) such that G(x) = Series_Reversion(F(x)) where F(x) = Integral 2/(2-x^2) dx = Sum_{n>=0} (2*n)!/2^n * x^(2*n+1)/(2*n+1)!.
From Vaclav Kotesovec, Jan 03 2014: (Start)
E.g.f.: -1+2/(1-sinh(sqrt(2)*x-arccosh(sqrt(2)))).
E.g.f.: (-1+exp(sqrt(2)*x)) * (3+2*sqrt(2)+exp(sqrt(2)*x)) / ((3+2*sqrt(2)-exp(sqrt(2)*x)) * (1+exp(sqrt(2)*x))).
a(n) ~ n! * 2^((n+1)/2) / (arccosh(3))^(n+1).
a(n) ~ n! * 2^((n+1)/2) / (log(3+2*sqrt(2)))^(n+1).
(End)
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 2*x^3/3! + 8*x^4/4! + 34*x^5/5! + 154*x^6/6! +...
Related series:
(1 + A(x))*sqrt(1 + A(x)^2) = 1 + x + 2*x^2/2! + 8*x^3/3! + 34*x^4/4! +...
sqrt(1 + A(x)^2) = 1 + x^2/2! + 3*x^3/3! + 8*x^4/4! + 30*x^5/5! + 154*x^6/6! +...
Let G(x) = 1 + A(x) - sqrt(1 + A(x)^2), an odd function that begins:
G(x) = x - x^3/3! + 4*x^5/5! - 34*x^7/7! + 496*x^9/9! - 11056*x^11/11! +...
then A(x) = G(x)*(2 - G(x))/(2 - 2*G(x)) such that
Series_Reversion(G(x)) = x + x^3/3! + 6*x^5/5! + 90*x^7/7! + 2520*x^9/9! + 113400*x^11/11! +...+ (2*n)!/2^n*x^(2*n+1)/(2*n+1)! +...
which equals Integral 2/(2-x^2) dx = log((1+x/sqrt(2))/(1-x/sqrt(2)))/sqrt(2).
MATHEMATICA
Rest[CoefficientList[Series[-1+2/(1-Sinh[Sqrt[2]*x-ArcCosh[Sqrt[2]]]), {x, 0, 10}], x] * Range[0, 10]!] (* Vaclav Kotesovec, Jan 03 2014 *)
PROG
(PARI) /* By definition, A(x) = sinh(x + Integral A(x) dx): */
{a(n)=local(A=x); for(i=1, n, A=sinh(x+intformal(A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) /* From A'(x) = (1 + A(x))*sqrt(1 + A(x)^2): */
{a(n)=local(A=x); for(i=1, n, A=intformal((1+A)*sqrt(1+A^2 +x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) /* A = G*(2-G)/(2-2*G), G = Series_Reversion(Integral 2/(2-x^2) dx): */
{a(n)=local(G=serreverse(intformal(2/(2-x^2 +x*O(x^n))))); n!*polcoeff(G*(2-G)/(2-2*G), n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 02 2014
STATUS
approved