OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(x * (exp(x) - 1)) ).
E.g.f.: A(x) = -x * (1 - exp(x))/LambertW(-x * (1 - exp(x))).
MATHEMATICA
nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(Exp[x] - 1)*x/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*(exp(x)-1)))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*(1-exp(x))/lambertw(-x*(1-exp(x)))))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 03 2022
STATUS
approved