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A134200 E.g.f. satisfies: A(x) = x*(exp(sinh(A(x)))).

%I A134200 #25 Nov 18 2024 22:32:20
%S A134200 0,1,2,9,68,725,9942,166453,3290632,75017097,1937420010,55906879809,
%T A134200 1782695466444,62247810769053,2362246665531326,96806321000599725,
%U A134200 4260677055123222544,200440759819510706321,10037364633737549049042,533071599267242747118585
%N A134200 E.g.f. satisfies: A(x) = x*(exp(sinh(A(x)))).
%H A134200 Alois P. Heinz, <a href="/A134200/b134200.txt">Table of n, a(n) for n = 0..300</a>
%F A134200 a(n) = sum(k=1..n-1, ((sum(i=0..k, (-1)^i*(k-2*i)^(n-1)* C(k,i))) *n^k)/(2^k*k!)), a(0)=0, a(1)=1. - _Vladimir Kruchinin_, May 10 2011
%F A134200 a(n) ~ n^(n-1) / (exp(n) * r^n * sqrt(1/s^2+sinh(s))), where r = 0.3296546568511367672... and s = 0.7650099545507321226... are roots of the system of equations exp(sinh(s))*r = s, s*cosh(s) = 1. - _Vaclav Kotesovec_, Jul 16 2014
%p A134200 A:= proc(n) option remember; if n=0 then 0 else convert (series (x* (exp (sinh(A(n-1)))), x=0, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n)*n!: seq (a(n), n=0..30);
%t A134200 CoefficientList[InverseSeries[Series[x/E^Sinh[x],{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jul 16 2014 *)
%o A134200 (Maxima) a(n):=if n<2 then n else sum(((sum((-1)^i*(k-2*i)^(n-1) *binomial(k,i), i,0,k))*n^k)/(2^k*k!),k,1,n-1); /* _Vladimir Kruchinin_, May 10 2011 */
%K A134200 nonn
%O A134200 0,3
%A A134200 _Alois P. Heinz_, Aug 27 2008