cp's OEIS Frontend

A226758 E.g.f.: A(x) = x + sin(A(x)^2).

%I A226758 #7 Jan 23 2014 10:02:09
%S A226758 1,2,12,120,1680,30120,658560,16994880,505612800,17037851040,
%T A226758 641393786880,26678131159680,1215016298496000,60135628841608320,
%U A226758 3213908573331456000,184463573184501811200,11316253482729190195200,738934748606732911833600,51171600229826941786521600
%N A226758 E.g.f.: A(x) = x + sin(A(x)^2).
%F A226758 E.g.f.: Series_Reversion(x - sin(x^2)).
%F A226758 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) sin(x^2)^n/n!.
%F A226758 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*sin(x^2)^n/n! ).
%F A226758 a(n) ~ n^(n-1) * sqrt(r/(1/s - 4*s^2*(s-r))) / (exp(n) * r^n), where s = 0.5186522338890123015... is the root of the equation 2*s*cos(s^2) = 1, and r = s - sin(s^2) = 0.2528845666082260013... - _Vaclav Kotesovec_, Jan 23 2014
%e A226758 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
%e A226758 where A(x - sin(x^2)) = x and A(x) = x + sin(A(x)^2).
%e A226758 Series expansions:
%e A226758 A(x) = x + sin(x^2) + d/dx sin(x^2)^2/2! + d^2/dx^2 sin(x^2)^3/3! + d^3/dx^3 sin(x^2)^4/4! +...
%e A226758 log(A(x)/x) = sin(x^2)/x + d/dx (sin(x^2)^2/x)/2! + d^2/dx^2 (sin(x^2)^3/x)/3! + d^3/dx^3 (sin(x^2)^4/x)/4! +...
%t A226758 Rest[CoefficientList[InverseSeries[Series[x - Sin[x^2],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 23 2014 *)
%o A226758 (PARI) {a(n)=n!*polcoeff(serreverse(x-sin(x^2+x^2*O(x^n))), n)}
%o A226758 for(n=1, 25, print1(a(n), ", "))
%o A226758 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A226758 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, sin(x^2+x*O(x^n))^m)/m!); n!*polcoeff(A, n)}
%o A226758 for(n=1, 25, print1(a(n), ", "))
%o A226758 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A226758 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, sin(x^2+x*O(x^n))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
%o A226758 for(n=1, 25, print1(a(n), ", "))
%Y A226758 Cf. A215188, A226759, A226760.
%K A226758 nonn
%O A226758 1,2
%A A226758 _Paul D. Hanna_, Jun 16 2013