A205806 E.g.f. A(x) satisfies: A( cos(x) - exp(-x) ) = x.

1, 2, 11, 100, 1269, 20680, 411655, 9681040, 262644825, 8074470560, 277410402675, 10533370203200, 438024379604525, 19798139730512000, 966408931916064975, 50666524133429152000, 2839464166814487200625, 169393547843598544960000, 10717798206604377757886875

Offset

1,2

Links

Table of n, a(n) for n=1..19.

Formula

a(n) ~ n^(n-1) / (sqrt(cos(s)+sin(s)) * exp(n) * (cos(s)-sin(s))^(n-1/2)), where s = 0.5885327439818610774... is the root of the equation sin(s) = exp(-s). - Vaclav Kotesovec, Jan 26 2014

Example

E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1269*x^5/5! +...
The series reversion of the e.g.f. begins:
cos(x) - exp(-x) = x - 2*x^2/2! + x^3/3! + x^5/5! - 2*x^6/6! + x^7/7! + x^9/9! - 2*x^10/10! + x^11/11! + x^13/13! - 2*x^14/14! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[Cos[x] - E^(-x), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 26 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(cos(x+x^2*O(x^n))-exp(-x+x^2*O(x^n))), n)}

Crossrefs

Sequence in context: A282640 A099169 A143135 * A220433 A318007 A243950

Adjacent sequences: A205803 A205804 A205805 * A205807 A205808 A205809

Keyword

nonn

Author

Paul D. Hanna, Feb 02 2012

Status

approved