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 A227461 E.g.f. equals the series reversion of x - x*arctanh(x). 1
 1, 2, 12, 128, 1920, 37104, 877184, 24520320, 791112960, 28932902400, 1182789053952, 53447706998784, 2645389044480000, 142326283714836480, 8270318699325112320, 516187815998727389184, 34440412737701955502080, 2446191865021002009477120, 184278436717136012676956160 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Note that arctanh(x) = log((1+x)/(1-x))/2. LINKS FORMULA E.g.f. A(x) satisfies: (1) A(x - x*arctanh(x)) = x. (2) A(x) = x/(1 - arctanh(A(x))). (3) A(x) = tanh( (A(x)-x)/A(x) ). (4) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * arctanh(x)^n / n!. (5) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * arctanh(x)^n / n! ). a(n) ~ n^(n-1) * s^2 * (1/s^2-1)^(n+1/2) / (exp(n) * sqrt(2)), where s = 0.43415423687337693781... is the root of the equation (1-s^2)*(1-arctanh(s)) = s. - Vaclav Kotesovec, Jan 13 2014 EXAMPLE E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! + ... where A(x) = x/(1 - arctanh(A(x))). The e.g.f. satisfies: (4) A(x) = x + x*arctanh(x) + d/dx x^2*arctanh(x)^2/2! + d^2/dx^2 x^3*arctanh(x)^3/3! + d^3/dx^3 x^4*arctanh(x)^4/4! + ... (5) log(A(x)/x) = arctanh(x) + d/dx x*arctanh(x)^2/2! + d^2/dx^2 x^2*arctanh(x)^3/3! + d^3/dx^3 x^3*arctanh(x)^4/4! + ... MATHEMATICA Rest[CoefficientList[InverseSeries[Series[x - x*ArcTanh[x], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *) PROG (PARI) {a(n)=n!*polcoeff(serreverse(x-x*atanh(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*atanh(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)*atanh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) CROSSREFS Cf. A227460. Sequence in context: A143136 A214224 A214431 * A228608 A097629 A259267 Adjacent sequences: A227458 A227459 A227460 * A227462 A227463 A227464 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 13 2013 STATUS approved

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Last modified January 27 11:39 EST 2023. Contains 359839 sequences. (Running on oeis4.)