%I #27 Mar 05 2024 11:28:30
%S 1,1,1,10,37,716,4741,136760,1314377,50468752,637409641,30580648352,
%T 479025538861,27578021183168,515932095998957,34657964676194176,
%U 754078761294069649,57902855910383448320,1436649321508321044817,124128617507138965088768,3459197142121422461242421
%N Expansion of e.g.f.: sqrt( -LambertW(-x) / LambertW(x) ).
%H G. C. Greubel, <a href="/A215881/b215881.txt">Table of n, a(n) for n = 0..385</a>
%F E.g.f.: exp( Sum_{n>=0} (2*n+1)^(2*n) * x^(2*n+1)/(2*n+1)! ).
%F a(n) = Sum_{k=0..n} -(-1)^k*C(n,k) * (k - 1/2)^(k-1) * (n-k + 1/2)^(n-k-1) / 4.
%F a(n) ~ c * n^(n-1), where c = 1/2*(1-LambertW(exp(-1))) / sqrt(LambertW(exp(-1))) = 0.6836640292259232... if n is even and c = 1/2*(1+LambertW(exp(-1))) / sqrt(LambertW(exp(-1))) = 1.2113614261884947... if n is odd. - _Vaclav Kotesovec_, Nov 27 2012
%e E.g.f.: A(x) = 1 + x + x^2/2! + 10*x^3/3! + 37*x^4/4! + 716*x^5/5! + 4741*x^6/6! +...
%e such that A(x) = sqrt( -LambertW(-x)/LambertW(x) ) where
%e LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! +...+ (-n)^(n-1)*x^n/n! +...
%e Related expansions:
%e log(A(x)) = x + 9*x^3/3! + 625*x^5/5! + 117649*x^7/7! + 43046721*x^9/9! +...+ (2*n-1)^(2*n-2)*x^(2*n-1)/(2*n-1)! +...
%t CoefficientList[Series[Sqrt[-LambertW[-x]/LambertW[x]], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)
%o (PARI) {a(n)=local(LW=sum(m=1, n+1,-(-1)^m*m^(m-1)*x^m/m!)+x^2*O(x^n)); n!*polcoeff(sqrt(-subst(LW, x, -x)/LW), n)}
%o (PARI) {a(n)=n!*polcoeff(exp(sum(m=0, n, (2*m+1)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)), n)}
%o (PARI) {a(n)=sum(k=0, n, -(-1)^k*binomial(n, k)*(k-1/2)^(k-1)*(n-k+1/2)^(n-k-1)/4)}
%o for(n=0,21,print1(a(n),", "))
%o (PARI) x='x+O('x^30); Vec(serlaplace(sqrt(-lambertw(-x)/lambertw(x)))) \\ _G. C. Greubel_, Feb 19 2018
%o (GAP) List([0..25],n->Sum([0..n],k->-(-1)^k*Binomial(n,k)*(k-(1/2))^(k-1)*(n-k+(1/2))^(n-k-1)/4)); # _Muniru A Asiru_, Feb 19 2018
%Y Cf. A215880, A215882, A215890, A138734, A216143.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Aug 25 2012