login
Expansion of e.g.f.: -LambertW(-x) / LambertW(x).
5

%I #31 Mar 12 2024 02:40:19

%S 1,2,4,26,160,2002,21184,395866,5980160,149083874,2933576704,

%T 91549564570,2222207205376,83345185392562,2407376957456384,

%U 105482963294851418,3534260251308064768,177194291803516980418,6757029862401745616896,381514700506253250858778

%N Expansion of e.g.f.: -LambertW(-x) / LambertW(x).

%H Vincenzo Librandi, <a href="/A215882/b215882.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: exp( Sum_{n>=0} 2*(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)^(k-1) * (n-k+1)^(n-k-1).

%F a(n) ~ c * n^(n-1), where c = (1-LambertW(exp(-1))^2)/LambertW(exp(-1)) = 3.31265693390754834... if n is even and c = (1+LambertW(exp(-1))^2)/ LambertW(exp(-1)) = 3.86958601942969593... if n is odd. - _Vaclav Kotesovec_, Nov 27 2012

%e E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 26*x^3/3! + 160*x^4/4! + 2002*x^5/5! +... such that A(x) = -LambertW(-x)/LambertW(x) where 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: log(A(x)) = 2*x + 18*x^3/3! + 1250*x^5/5! + 235298*x^7/7! + 86093442*x^9/9! +...+ 2*(2*n+1)^(2*n)*x^(2*n+1)/(2*n+1)! +...

%t CoefficientList[Series[-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*(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)^(k-1)*(n-k+1)^(n-k-1))}

%o for(n=0,21,print1(a(n),", "))

%o (PARI) x='x+O('x^30); Vec(serlaplace(-lambertw(-x)/lambertw(x))) \\ _G. C. Greubel_, Feb 19 2018

%o (GAP) List([0..20],n->Sum([0..n],k->-(-1)^k*Binomial(n,k)*(k-1)^(k-1)*(n-k+1)^(n-k-1))); # _Muniru A Asiru_, Feb 20 2018

%Y Cf. A215880, A215881, A138737, A215890.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 25 2012