A224080 E.g.f. is series reversion of log(1+x)*sqrt(1-x^2).

1, 1, 4, 25, 231, 2701, 38963, 662929, 13037608, 290586301, 7241964357, 199511192233, 6020966627261, 197526938563501, 6999280556977816, 266408120037084577, 10840080711977589375, 469561023814159909981, 21573682777922810043335, 1047866254345761285979321

Offset

1,3

Comments

Compare to: Series_Reversion(log(1+x)*(1+x)) = Sum_{n>=1} -(n-1)^(n-1)*(-x)^n/n!.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

E.g.f. A(x) satisfies: 1 + A(x) = exp( x / sqrt(1 - A(x)^2) ).

a(n) ~ n^(n-1) * (s/((1-s)*sqrt(1-s^2)))^n * ((1-s^2)/sqrt(1+s+s^2)) / exp(n), where s = 0.66288612806066020129... is the root of the equation s*log(1+s) = 1-s. - Vaclav Kotesovec, Jan 13 2014

Example

E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 231*x^5/5! + 2701*x^6/6! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[Log[1+x]*Sqrt[1-x^2], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(serreverse(log(1+X)*sqrt(1-X^2)), n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A121660 A118835 A365053 * A194569 A356000 A198058

Adjacent sequences: A224077 A224078 A224079 * A224081 A224082 A224083

Keyword

nonn

Author

Paul D. Hanna, Jul 20 2013

Status

approved