A133892 - OEIS

%I #14 Jul 16 2014 17:08:26

%S 0,1,2,6,36,360,4542,68544,1226792,25441920,598142170,15713984000,

%T 456391238028,14521095333888,502259604707798,18763725111828480,

%U 752970270575818192,32301914469949407232,1475208429063535282482,71458043399471257288704,3659301012147404374818420

%N E.g.f. satisfies: A(x) = x*(tan(sinh(A(x)))+1).

%H Alois P. Heinz, <a href="/A133892/b133892.txt">Table of n, a(n) for n = 0..150</a>

%F a(n) ~ n^(n-1) * sqrt(s/((2*r^2-2*r*s+s^2)*(2*(s-r)*(cosh(s))^2 + r*sinh(s)))) / (exp(n) * r^(n-1)), where r = 0.3611557635751972926... and s = 0.7471844338670576493... are roots of the system of equations r*cosh(s) = (cos(sinh(s)))^2, s = r + r*tan(sinh(s)). - _Vaclav Kotesovec_, Jul 16 2014

%p A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (tan (sinh(A(n-1)))+1), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..25);

%p # second Maple program:

%p a:= n-> n!*coeff(series(RootOf(A=x*(tan(sinh(A))+1), A), x, n+1), x, n):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Mar 05 2013

%t CoefficientList[InverseSeries[Series[x/(1 + Tan[Sinh[x]]),{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jul 16 2014 *)

%Y Cf. A133822.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 27 2008