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A200560 E.g.f.: arcsin(x) o x/(1-x) o sin(x), a composition of functions involving sin(x) and its inverse.

%I A200560 #24 Apr 05 2016 07:40:09
%S A200560 1,2,6,28,180,1502,15456,189208,2683920,43263962,780807456,
%T A200560 15593180788,341340941760,8126644655222,209050212857856,
%U A200560 5777935570510768,170755837008595200,5373097909706399282,179351443518333574656,6329687401322560131148,235491796312126982538240
%N A200560 E.g.f.: arcsin(x) o x/(1-x) o sin(x), a composition of functions involving sin(x) and its inverse.
%C A200560 Given e.g.f. A(x), then A(Pi/6) = Pi/2, where Pi/6 is the radius of convergence.
%H A200560 Vaclav Kotesovec, <a href="/A200560/b200560.txt">Table of n, a(n) for n = 1..250</a>
%F A200560 E.g.f. A(x) satisfies: A(-A(-x)) = x.
%F A200560 The n-th iteration of e.g.f. A(x) equals: arcsin(x) o x/(1-n*x) o sin(x) = arcsin( sin(x)/(1-n*sin(x)) ).
%F A200560 a(n) ~ 2^n * 3^(n-1/4) * n^(n-1) / (Pi^(n-1/2) * exp(n)). - _Vaclav Kotesovec_, Apr 05 2016
%e A200560 E.g.f.: A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! +...
%e A200560 where the initial iterations of e.g.f. A(x) begin:
%e A200560 A(A(x)) = arcsin( sin(x)/(1-2*sin(x)) ); more explicitly,
%e A200560 A(A(x)) = x + 4*x^2/2! + 24*x^3/3! + 200*x^4/4! + 2160*x^5/5! +...
%e A200560 A(A(A(x))) = arcsin( sin(x)/(1-3*sin(x)) ); more explicitly,
%e A200560 A(A(A(x))) = x + 6*x^2/2! + 54*x^3/3! + 660*x^4/4! + 10260*x^5/5! +...
%e A200560 A(A(A(A(x)))) = arcsin( sin(x)/(1-4*sin(x)) ); more explicitly,
%e A200560 A(A(A(A(x)))) = x + 8*x^2/2! + 96*x^3/3! + 1552*x^4/4! + 31680*x^5/5! +...
%o A200560 (PARI) {a(n)=n!*polcoeff(subst(asin(x+x*O(x^n)), x, subst(x/(1-x), x, sin(x+x*O(x^n)))), n)}
%K A200560 nonn
%O A200560 1,2
%A A200560 _Paul D. Hanna_, Nov 29 2011