1,3

Also, one third of the number of proper vertex colorings of the n-complete tripartite graph using exactly 5 interchangeable colors.

The complete 3-partite graph K(n,n,n) has 3n vertices partitioned into three sets of size n each, with edges between every pair of vertices from different sets. 3*a(n) = 0 for n < 2 because we need at least 2 vertices per partition to create 5 nonempty independent sets.

Vincenzo Librandi, Table of n, a(n) for n = 1..500

Julian Allagan, Gabrielle Morgan, and Deonna Sinclair, Bell Numbers and Stirling Numbers of the Mycielskian of Trees, arXiv:2512.06980 [math.CO], 2025. See pp. 1, 8, 10-11.

Richard P. Stanley, Enumerative Combinatorics, Cambridge University Press.

Eric Weisstein's World of Mathematics, Complete Multipartite Graph.

Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind.

Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

3*a(n) = 2^(2*n - 2) + (1/2)*3^(n - 1) - 3*2^(n - 1) + 3/2 for n >= 1.

G.f.: 1/(4*(1 - 4*x)) + 1/(6*(1 - 3*x)) - 3/(2*(1 - 2*x)) + 3/(2*(1 - x)).

a(n) = A385432(n, 5) / 3 = A060867(n-1) + A000392(n).

From Stefano Spezia, Jun 14 2025: (Start)

a(n) = (6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4.

E.g.f.: (exp(x) - 1)^2*(3*exp(2*x) + 8*exp(x) - 5)/12. (End)

a(n) = A000453(n+2) -10*A000453(n). - R. J. Mathar, Jul 20 2025

3*a(2) = 3 because K(2,2,2) can be partitioned into 5 nonempty independent sets in exactly 3 ways.

Table[(StirlingS2[n, 3] + StirlingS2[n, 2]^2), {n, 1, 20}]

(Magma) [(6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4: n in [1..30]]; // Vincenzo Librandi, Jul 24 2025

Cf. A000392, A000453, A060867, A385432.

Sequence in context: A022575 A202481 A348663 * A169720 A024210 A253002

Adjacent sequences: A384985 A384986 A384987 * A384989 A384990 A384991

nonn,easy

Julian Allagan, Jun 14 2025

approved