OFFSET
0,4
COMMENTS
This sequence gives the number of unordered pairs {A, B} of disjoint nonempty subsets of an n-element set such that A and B are nonempty, and not (|A| = 1 and |B| = 1). Correspondingly, it counts all pairs of disjoint nonempty subsets excluding the case where both subsets are singletons. This provides a combinatorial interpretation of a sequence previously defined via polylogarithmic expressions (see A142249). - Md. Rad Sarar Anando, Mar 28 2026
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (8,-24,34,-23,6).
FORMULA
a(n) ~ 3^n / 2.
From Md. Rad Sarar Anando, Mar 28 2026: (Start)
O.g.f.: x^3*(3 - 5*x)/((1 - x)^3*(1 - 2*x)(1 - 3*x)).
a(n) = Stirling2(n+1, 3) - binomial(n, 2).
a(n) = (3^n - 2^(n+1) + 1 - n^2 + n)/2.
a(n) = a(n-1) + 3^(n-1) - 2^(n-1) - n + 1, for n >= 4, with a(3) = 3. (End)
MATHEMATICA
Join[{0, 0, 0}, Table[(CoefficientList[Simplify[(1-x)^n * PolyLog[-n, 2, x] / (x*Log[1-x])], x]/.x->1-E)[[3]], {n, 4, 20}]]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Oct 12 2017
EXTENSIONS
Name supplemented with a formula by Md. Rad Sarar Anando by Peter Luschny, Mar 28 2026
Offset set to 0 and first terms added by Peter Luschny, Mar 29 2026
STATUS
approved