OFFSET
0,4
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
FORMULA
E.g.f.: (exp(x)-1)^3 + (exp(x)-1)^2/2!.
From Colin Barker, Jan 30 2017: (Start)
G.f.: x^2*(1 + 3*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3.
a(n) = 2 - 5*2^(n-1) + 3^n for n>0. (End)
EXAMPLE
a(3) = 9 because we have: {{}, {c}, {a,b}, {a,b,c}} with 3 labelings and {{}, {c}, {b,c}, {a,b,c}} with 6 labelings.
MATHEMATICA
CoefficientList[Series[x^2*(1 + 3 x)/((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 27}], x] (* Michael De Vlieger, Jan 21 2018 *)
PROG
(PARI) a(n) = stirling(n, 2, 2) + 3!*stirling(n, 3, 2) \\ Colin Barker, Jan 30 2017
(PARI) concat(vector(2), Vec(x^2*(1 + 3*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017
STATUS
approved