A003468 Number of minimal 3-covers of a labeled n-set.
1, 22, 305, 3410, 33621, 305382, 2619625, 21554170, 171870941, 1337764142, 10216988145, 76862115330, 571247591461, 4203844925302, 30687029023865, 222518183370890, 1604626924403181, 11518132293452862
Offset: 3
Keywords
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 3..1000
- T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Eric Weisstein's World of Mathematics, Minimal cover.
- Index entries for linear recurrences with constant coefficients, signature (22, -179, 638, -840).
Crossrefs
Programs
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Magma
[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6: n in [3..30]]; // Vincenzo Librandi, May 03 2013
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Maple
A003468:=1/(6*z-1)/(4*z-1)/(7*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation
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Mathematica
Table[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6, {n, 3, 20}] (* Vaclav Kotesovec, Nov 19 2012 *) LinearRecurrence[{22,-179,638,-840},{1,22,305,3410},20] (* Harvey P. Dale, Jan 09 2024 *)
Formula
G.f.: x^3/((1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)). - N. J. A. Sloane, May 12 1994, corrected by Vaclav Kotesovec, Nov 19 2012
E.g.f.: (exp(4*x)*(exp(x) - 1)^3)/6. More generally, e.g.f. for number of minimal m-covers of a labeled n-set is (exp((2^m - m - 1)*x)*(exp(x) - 1)^m)/m!. - Vladeta Jovovic, May 09 2004
If we define f(m, j, x) = sum(binomial(m, k)*stirling2(k, j)*x^(m - k),k = j .. m) then a(n) = f(n, 3, 4), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 7^n/6 - 6^n/2 + 5^n/2 - 4^n/6. - Vaclav Kotesovec, Nov 19 2012
Comments