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 A227460 E.g.f. equals the series reversion of x - x*arctan(x). 1
 1, 2, 12, 112, 1440, 23664, 473984, 11203200, 305268480, 9421739520, 324874716672, 12377880797184, 516412532213760, 23414906122352640, 1146462496418119680, 60286544709800361984, 3388531521749583790080, 202734601173679682027520, 12863997904326532359782400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS FORMULA E.g.f. A(x) satisfies: (1) A(x - x*arctan(x)) = x. (2) A(x) = x/(1 - arctan(A(x))). (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * arctan(x)^n / n!. (4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * arctan(x)^n / n! ). a(n) ~ n^(n-1) * s^2 * (1+1/s^2)^(n+1/2) / (sqrt(2)*exp(n)), where s = 0.61728206508912310932... is the root of the equation (1+s^2)*(1-arctan(s)) = s. - Vaclav Kotesovec, Jan 13 2014 EXAMPLE E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + ... where A(x) = x/(1 - arctan(A(x))). The e.g.f. satisfies: (3) A(x) = x + x*arctan(x) + d/dx x^2*arctan(x)^2/2! + d^2/dx^2 x^3*arctan(x)^3/3! + d^3/dx^3 x^4*arctan(x)^4/4! + ... (4) log(A(x)/x) = arctan(x) + d/dx x*arctan(x)^2/2! + d^2/dx^2 x^2*arctan(x)^3/3! + d^3/dx^3 x^3*arctan(x)^4/4! + ... MATHEMATICA Rest[CoefficientList[InverseSeries[Series[x - x*ArcTan[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *) PROG (PARI) {a(n)=n!*polcoeff(serreverse(x-x*atan(x +x*O(x^n))), n)} for(n=1, 25, print1(a(n), ", ")) (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*atan(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*atan(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) CROSSREFS Cf. A227461. Sequence in context: A143134 A214225 A185190 * A316651 A330654 A091481 Adjacent sequences:  A227457 A227458 A227459 * A227461 A227462 A227463 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 13 2013 STATUS approved

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Last modified April 4 05:34 EDT 2020. Contains 333212 sequences. (Running on oeis4.)