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

1,2

Comments

Compare e.g.f. to the identity: cosh(x) = exp(x) - sqrt(cosh(x)^2 - 1).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

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

Example

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! +...

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

Mathematica

Rest[CoefficientList[Series[(E^x-Sqrt[2-E^(2*x)])/2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 22 2013 *)

Prog

(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)}

Crossrefs

Cf. A194349.

Partial sums of A352117.

Sequence in context: A196793 A306880 A128579 * A346258 A242105 A001046

Adjacent sequences: A194450 A194451 A194452 * A194454 A194455 A194456

Keyword

nonn

Author

Paul D. Hanna, Aug 24 2011

Status

approved