A194453 E.g.f. satisfies: A(x) = exp(x) - sqrt(1 - A(x)^2).
1, 2, 7, 44, 421, 5342, 83707, 1556984, 33495721, 817880282, 22341817807, 675009140324, 22347321835021, 804481291160822, 31286388389010307, 1307157133950142064, 58390601701376026321, 2776992745284738150962, 140092142842449580093207
Offset: 1
Keywords
Examples
E.g.f.: A(x) = x + 2*x^2/2! + 7*x^3/3! + 44*x^4/4! + 421*x^5/5! +... where A( log(sqrt(1-x^2) + x) ) = x and log(sqrt(1-x^2) + x) = x - 2*x^2/2! + 5*x^3/3! - 24*x^4/4! + 129*x^5/5! - 960*x^6/6! +...+ -(-1)^n*A194349(n)*x^n/n! +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
Programs
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Maple
a:= n-> n!*coeff(series(RootOf(A=exp(x)-sqrt(1-A^2), A), x, n+1), x, n): seq(a(n), n=1..20); # Alois P. Heinz, Sep 21 2013
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Mathematica
Rest[CoefficientList[Series[(E^x-Sqrt[2-E^(2*x)])/2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 22 2013 *) -
PARI
{a(n)=n!*polcoeff(serreverse( log(sqrt(1-x^2 +O(x^(n+2)))+x)),n)} -
PARI
{a(n)=n!*polcoeff((exp(x+x*O(x^n))-sqrt(2-exp(2*x+x*O(x^n))))/2,n)}
Formula
E.g.f.: A(x) = (exp(x) - sqrt(2 - exp(2*x))) / 2.
E.g.f. A(x) satisfies: A( log(sqrt(1-x^2) + x) ) = x; thus, e.g.f. A(x) is a signed series reversion of the e.g.f. of A194349.
E.g.f. A(x) satisfies: A(x) = sinh(x) + exp(-x)*A(x)^2. - Paul D. Hanna, Aug 29 2018
a(n) ~ (2*n)^(n-1) / (exp(n) * log(2)^(n-1/2)). - Vaclav Kotesovec, Sep 22 2013
Comments