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A035348
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Triangle of a(n,k) = number of k-member minimal covers of an n-set (n >= 1, k >= 1).
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9
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1, 1, 1, 1, 6, 1, 1, 25, 22, 1, 1, 90, 305, 65, 1, 1, 301, 3410, 2540, 171, 1, 1, 966, 33621, 77350, 17066, 420, 1, 1, 3025, 305382, 2022951, 1298346, 100814, 988, 1, 1, 9330, 2619625, 47708115, 83384427, 18151560, 549102, 2259, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| These are what Clarke calls "Minimal disordered k-covers of labeled n-set".
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REFERENCES
| R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
Macula, Anthony J., Lewis Carroll and the enumeration of minimal covers. Math. Mag. 68 (1995), 269-274.
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LINKS
| Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| a(n,k) = Sum_{j >= 0} (-1)^j * binomial(k,j) * (2^k-1-j)^n. [Hearne-Wagner]
a(n,k) = Sum_{j >= k} binomial(2^k-k-1,j-k)*j!*Stirling2(n,j). [Macula]
E.g.f.: Sum((exp(y)-1)^n*exp(y*(2^n-n-1))*x^n/n!, n=0..infinity). - Vladeta Jovovic (vladeta(AT)eunet.rs), May 08 2004
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EXAMPLE
| 1; 1,1; 1,6,1; 1,25,22,1; ...
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MATHEMATICA
| a[n_, k_] := Sum[ (-1)^i*(2^k-i-1)^n / (i!*(k-i)!), {i, 0, k}]; Flatten[ Table[ a[n, k], {n, 1, 9}, {k, 1, n}]] (* From Jean-François Alcover, Dec 13 2011, after Pari *)
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PROG
| (PARI) C(n, k) = if(k<0|k>n, 0, n!/k!/(n-k)!); a(n, k)=sum(i=0, k, (-1)^i*C(k, i)*(2^k-1-i)^n)/k!; printp(matrix(10, 10, n, k, a(n-1, k-1)))
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CROSSREFS
| Row sums are A046165. Cf. A049055, A003465, A002177.
Sequence in context: A173882 A174045 A169660 * A140945 A141688 A166960
Adjacent sequences: A035345 A035346 A035347 * A035349 A035350 A035351
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KEYWORD
| nonn,tabl,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Entry improved by Michael Somos
Explicit formulas added by N. J. A. Sloane, Aug 05 2011
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