A384988 a(n) = Stirling2(n,2)^2 + Stirling2(n,3).
0, 1, 10, 55, 250, 1051, 4270, 17095, 68050, 270451, 1075030, 4276735, 17030650, 67881451, 270777790, 1080817975, 4316294050, 17244046051, 68912400550, 275457464815, 1101251874250, 4403270396251, 17607863991310, 70415790601255, 281616141147250, 1126323450484051
Offset: 1
Examples
3*a(2) = 3 because K(2,2,2) can be partitioned into 5 nonempty independent sets in exactly 3 ways.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..500
- 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).
Programs
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Magma
[(6 - 3*2^(n+1) + 2*3^(n-1) + 4^n)/4: n in [1..30]]; // Vincenzo Librandi, Jul 24 2025
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Mathematica
Table[(StirlingS2[n, 3] + StirlingS2[n, 2]^2), {n, 1, 20}]
Formula
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)).
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)
Comments