A003468 - OEIS

%I M5125 #66 Jan 09 2024 11:45:40

%S 1,22,305,3410,33621,305382,2619625,21554170,171870941,1337764142,

%T 10216988145,76862115330,571247591461,4203844925302,30687029023865,

%U 222518183370890,1604626924403181,11518132293452862

%N Number of minimal 3-covers of a labeled n-set.

%C This is also the fourth column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See the e.g.f. given below. See also the Sheffer comments in A193685. - _Wolfdieter Lang_, Oct 08 2011

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A003468/b003468.txt">Table of n, a(n) for n = 3..1000</a>

%H T. Hearne and C. G. Wagner, <a href="http://dx.doi.org/10.1016/0012-365X(73)90141-6">Minimal covers of finite sets</a>, Discr. Math. 5 (1973), 247-251.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MinimalCover.html">Minimal cover.</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (22, -179, 638, -840).

%F 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

%F 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

%F 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

%F a(n) = 7^n/6 - 6^n/2 + 5^n/2 - 4^n/6. - _Vaclav Kotesovec_, Nov 19 2012

%p A003468:=1/(6*z-1)/(4*z-1)/(7*z-1)/(5*z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t Table[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6, {n, 3, 20}] (* _Vaclav Kotesovec_, Nov 19 2012 *)

%t LinearRecurrence[{22,-179,638,-840},{1,22,305,3410},20] (* _Harvey P. Dale_, Jan 09 2024 *)

%o (Magma) [7^n/6 - 6^n/2 + 5^n/2 - 4^n/6: n in [3..30]]; // _Vincenzo Librandi_, May 03 2013

%Y Cf. A000392, A003468, A016111, A046166-A046169, A057668, A005783-A005786, A055066.

%Y Cf. A143496, A000302, A005060, A016103.

%K nonn

%O 3,2

%A _N. J. A. Sloane_