OFFSET
0,3
FORMULA
E.g.f. satisfies:
(1) A(x) = (1/x)*Series_Reversion(x-x^2*cosh(x)).
(2) A(x) = 1/(1 - x*A(x)*cosh(x*A(x))).
(3) A(x-x^2*cosh(x)) = 1/(1-x*cosh(x)).
a(n) = [x^n/n!] 1/(1 - x*cosh(x))^(n+1) / (n+1).
a(n) ~ n^(n-1) * s*sqrt(1/(6-2*s-r^2*s^2+r^2*s^3)) / (exp(n) * r^n), where r = 0.2278231894714399793... and s = 1.855593992316816009... are the roots of the equations r*s*(2*cosh(r*s) + r*s*sinh(r*s)) = 1, 1 + r*s^2*cosh(r*s) = s. - Vaclav Kotesovec, Jan 13 2014
EXAMPLE
E.g.f: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 408*x^4/4! + 6725*x^5/5! +...
MATHEMATICA
CoefficientList[1/x*InverseSeries[Series[x-x^2*Cosh[x], {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 13 2014 *)
PROG
(PARI) {a(n)=n!*polcoeff(1/(1 - x*cosh(x+x*O(x^n)))^(n+1)/(n+1), n)}
(PARI) {a(n)=n!*polcoeff((1/x)*serreverse(x-x^2*cosh(x+x*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Aug 08 2012
STATUS
approved