OFFSET
0,2
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = Sum_{k=0..n} 2^k * (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (2 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(2 * (exp(x) - 1)) ).
E.g.f.: A(x) = 2 * (exp(x) - 1)/LambertW(2 * (exp(x) - 1)).
MATHEMATICA
nmax = 18; A[_] = 1;
Do[A[x_] = Exp[(2*(Exp[x] - 1))/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
PROG
(PARI) a(n) = sum(k=0, n, 2^k*(-k+1)^(k-1)*stirling(n, k, 2));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(2*(exp(x)-1))^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(exp(x)-1)))))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(2*(exp(x)-1)/lambertw(2*(exp(x)-1))))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Sep 19 2022
STATUS
approved