OFFSET
0,4
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = Sum_{k=0..n} 2^(n-k) * (-k+1)^(k-1) * |Stirling1(n,k)|.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-log(1-2*x)/2)^k / k!.
E.g.f.: A(x) = exp( LambertW(-log(1-2*x)/2) ).
E.g.f.: A(x) = -log(1-2*x)/(2 * LambertW(-log(1-2*x)/2)).
MATHEMATICA
nmax = 20; A[_] = 1;
Do[A[x_] = (1 - 2*x)^(-(1/2)/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) = sum(k=0, n, 2^(n-k)*(-k+1)^(k-1)*abs(stirling(n, k, 1)));
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-log(1-2*x)/2)^k/k!)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-log(1-2*x)/2))))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-log(1-2*x)/(2*lambertw(-log(1-2*x)/2))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 03 2022
STATUS
approved