OFFSET
0,5
COMMENTS
FORMULA
E.g.f.: exp(x)*(-3 + (1+x)*T^3/6 + x*T^2/2 + 3*x*T/2 + 3*x*(1+1/T)) where T = -LambertW(-x).
G.f.: Sum_{k>=1} (k-1)*(k-2)/2*k^(k-3)*x^k/(1-x)^(k+2).
Limit_{n->oo} a(n) / (n^(n-1)) = exp(exp(-1))/2.
MAPLE
T := -LambertW(-x): a := exp(x)*(-3 + (1+x)*T^3/6 + x*T^2/2 + 3*x*T/2 + 3*x*(1+1/T)):
ser := series(a, x = 0, 22): seq(n!*coeff(ser, x, n), n = 0 .. 20);
PROG
(Python)
from math import comb
def a(n):
return sum(comb(n+1, k+1)*comb(k-1, 2)*k**(k-3) for k in range(3, n+1))
print([a(n) for n in range(21)])
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mélika Tebni, Mar 08 2026
STATUS
approved
