A216401 - OEIS

%I #17 Aug 17 2018 21:31:26

%S 1,2,5,28,209,1806,18997,235544,3322881,52688890,929147141,

%T 18023207412,381330466321,8740727309510,215767934510325,

%U 5706703994412976,160994795504231297,4825786400923162482,153160894479441852037,5131078462229088189260

%N E.g.f.: arctanh(x*exp(x)).

%H Vincenzo Librandi, <a href="/A216401/b216401.txt">Table of n, a(n) for n = 1..200</a>

%F E.g.f.: Sum_{n>=1} exp((2*n-1)*x) * x^(2*n-1) / (2*n-1).

%F E.g.f.: log( (1+x*exp(x)) / (1-x*exp(x)) ) / 2.

%F a(n) = n! * Sum_{k=0..floor((n-1)/2)} (2*k+1)^(n-2*k-2) / (n-2*k-1)!.

%F a(n) ~ (n-1)!/(2*LambertW(1)^n). - _Vaclav Kotesovec_, Feb 12 2013

%e 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 such that tanh(A(x)) = x*exp(x) and

%e A(x) = exp(x)*x + exp(3*x)*x^3/3 + exp(5*x)*x^5/5 + exp(7*x)*x^7/7 + ...

%t 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 (PARI) {a(n)=n!*polcoeff(atanh(x*exp(x +x*O(x^n))),n)}

%o (PARI) {a(n)=n!*sum(k=0,(n-1)\2,(2*k+1)^(n-2*k-2)/(n-2*k-1)!)}

%o for(n=1,25,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 06 2012