OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( -LambertW(2 * x * (1 - exp(x)))/2 ).
E.g.f.: A(x) = ( LambertW(2 * x * (1 - exp(x)))/(2 * x * (1 - exp(x))) )^(1/2).
MATHEMATICA
m = 20; (* number of terms *)
CoefficientList[Exp[-(1/2)*LambertW[-2*(Exp[x]-1)*x]] + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Sep 11 2022 *)
PROG
(PARI) a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*(1-exp(x)))/2)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*(1-exp(x)))/(2*x*(1-exp(x))))^(1/2)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 28 2022
STATUS
approved