OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..336
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * Stirling1(n,k).
E.g.f.: A(x) = -Sum_{k>=0} (3*k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = ( -3*log(1-x)/LambertW(-3*log(1-x)) )^(1/3).
a(n) ~ -(-1)^n * exp(1/6 - exp(-1)/6 - n) * n^(n-1) / (sqrt(3) * (-1 + exp(exp(-1)/3))^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021
MATHEMATICA
nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1 - x))^(1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*stirling(n, k, 1));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (3*k-1)^(k-1)*log(1-x)^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-3*log(1-x)/lambertw(-3*log(1-x)))^(1/3)))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Nov 23 2021
STATUS
approved