A046166 -id:A046166 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

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

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).

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

Cf. A143496, A000302, A005060, A016103.

[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6: n in [3..30]]; // Vincenzo Librandi, May 03 2013

A003468:=1/(6*z-1)/(4*z-1)/(7*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

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 *)

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

A057966 Triangle T(n,k) of number of minimal 5-covers of a labeled n-set that cover k points of that set uniquely (k=5,..,n).

Original entry on oeis.org

1, 156, 15, 14196, 2730, 140, 984256, 283920, 29120, 1050, 57578976, 22145760, 3407040, 245700, 6951, 2994106752, 1439474400, 295276800, 31941000, 1807260, 42525, 142719088512, 82337935680, 21112291200, 3045042000, 258438180

Offset: 5

Vladeta Jovovic, Oct 17 2000

Row sums give A046166.

			[1], [156, 15], [14196, 2730, 140], [984256, 283920, 29120, 1050], ...; there are 15 minimal 5-covers of a labeled 6-set that cover 6 points of that set uniquely.

Eric Weisstein's World of Mathematics, Minimal cover

Cf. A035347, A057669, A057963-A057965, A057967, A057968(unlabeled case).

Number of minimal m-covers of a labeled n-set that cover k points of that set uniquely is C(n, k)*S(k, m)*(2^m-m-1)^(n-k), where S(k, m) are Stirling numbers of the second kind.

A057668 Number of minimal 7-covers of a labeled n-set.

Original entry on oeis.org

1, 988, 549102, 226064280, 76785889587, 22762819040676, 6092115565691584, 1505097773271664000, 348617485585838373333, 76564317282173987801964, 16080209472530744351164146, 3250906483045575317042337960, 635954979082842132795003641239

Offset: 7

Vladeta Jovovic, Oct 16 2000

Eric Weisstein's World of Mathematics, Minimal cover

Cf. A000392, A003468, A016111, A046166-A046169, A005783-A005786, A055066 (unlabeled case).

a(n) = (1/7!) * (127^n - 7 * 126^n + 21 * 125^n - 35 * 124^n + 35 * 123^n - 21 * 122^n + 7 * 121^n - 120^n).

G.f.: x^7 / ((120*x-1)*(121*x-1)*(122*x-1)*(123*x-1)*(124*x-1)*(125*x-1)*(126*x-1)*(127*x-1)). - Colin Barker, Jul 11 2013

Additional term from Colin Barker, Jul 11 2013

cp's OEIS Frontend

A003468 Number of minimal 3-covers of a labeled n-set.

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A057966 Triangle T(n,k) of number of minimal 5-covers of a labeled n-set that cover k points of that set uniquely (k=5,..,n).

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A057668 Number of minimal 7-covers of a labeled n-set.

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