OFFSET
1,2
FORMULA
E.g.f.: -LambertW(-x) / (1 - LambertW(-x)^2).
a(n) = n * A133297(n).
a(n) = n^n * hypergeom([1, 1-n], [], 1/n). - Peter Luschny, Mar 09 2026
MAPLE
a := n -> ifelse(n = 0, 0, n^n * hypergeom([1, 1-n], [], 1/n)):
seq(simplify(a(n)), n = 1..20); # Peter Luschny, Mar 09 2026
MATHEMATICA
Table[n! Sum[(-1)^(k + 1) n^(n - k)/(n - k)!, {k, 1, n}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x]/(1 - LambertW[-x]^2), {x, 0, nmax}], x] Range[0, nmax]! // Rest
PROG
(PARI) a(n) = n! * sum(k=1, n, (-1)^(k+1)*n^(n-k)/(n-k)!); \\ Michel Marcus, Sep 23 2021
(Python)
from math import factorial
def a(n: int) -> int:
if n == 0: return 0
t = factorial(n)
if n % 2 == 0: t = -t
f = t
for k in range(n, 1, -1):
t = -n * t // (n - k + 1)
f += t
return f
print([a(n) for n in range(1, 21)]) # Peter Luschny, Mar 09 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 23 2021
STATUS
approved