cp's OEIS Frontend

A216401 E.g.f.: artanh(x*exp(x)).

%I A216401 #21 Apr 16 2025 10:24:59
%S A216401 1,2,5,28,209,1806,18997,235544,3322881,52688890,929147141,
%T A216401 18023207412,381330466321,8740727309510,215767934510325,
%U A216401 5706703994412976,160994795504231297,4825786400923162482,153160894479441852037,5131078462229088189260
%N A216401 E.g.f.: artanh(x*exp(x)).
%H A216401 Vincenzo Librandi, <a href="/A216401/b216401.txt">Table of n, a(n) for n = 1..200</a>
%H A216401 Wikipedia contributors, <a href="https://en.wikipedia.org/wiki/Area_function_(inverse_hyperbolic_function)">Area function (inverse hyperbolic function)</a>, Wikipedia, the free encyclopedia, as of April 7, 2025.
%F A216401 E.g.f.: Sum_{n>=1} exp((2*n-1)*x) * x^(2*n-1) / (2*n-1).
%F A216401 E.g.f.: log( (1+x*exp(x)) / (1-x*exp(x)) ) / 2.
%F A216401 a(n) = n! * Sum_{k=0..floor((n-1)/2)} (2*k+1)^(n-2*k-2) / (n-2*k-1)!.
%F A216401 a(n) ~ (n-1)!/(2*LambertW(1)^n). - _Vaclav Kotesovec_, Feb 12 2013
%e A216401 E.g.f.: A(x) = x + 2*x^2/2! + 5*x^3/3! + 28*x^4/4! + 209*x^5/5! + 1806*x^6/6! + ...
%e A216401 such that tanh(A(x)) = x*exp(x) and
%e A216401 A(x) = exp(x)*x + exp(3*x)*x^3/3 + exp(5*x)*x^5/5 + exp(7*x)*x^7/7 + ...
%t A216401 CoefficientList[Series[Log[(1+x*E^x)/(1-x*E^x)]/2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Feb 12 2013 *)
%o A216401 (PARI) {a(n)=n!*polcoeff(atanh(x*exp(x +x*O(x^n))),n)}
%o A216401 (PARI) {a(n)=n!*sum(k=0,(n-1)\2,(2*k+1)^(n-2*k-2)/(n-2*k-1)!)}
%o A216401 for(n=1,25,print1(a(n),", "))
%K A216401 nonn
%O A216401 1,2
%A A216401 _Paul D. Hanna_, Sep 06 2012