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A057963
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Triangle T(n,k) of number of minimal 2-covers of a labeled n-set that cover k points of that set uniquely (k=2,..,n).
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12
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1, 3, 3, 6, 12, 7, 10, 30, 35, 15, 15, 60, 105, 90, 31, 21, 105, 245, 315, 217, 63, 28, 168, 490, 840, 868, 504, 127, 36, 252, 882, 1890, 2604, 2268, 1143, 255, 45, 360, 1470, 3780, 6510, 7560, 5715, 2550, 511, 55, 495, 2310, 6930, 14322, 20790, 20955, 14025
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OFFSET
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2,2
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COMMENTS
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Row sums give A000392.
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LINKS
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Robert Israel, Table of n, a(n) for n = 2..10012 (rows 2 to 142, flattened)
Octavio Alberto Agustín-Aquino, Archimedes' quadrature of the parabola and minimal covers, arXiv:1602.05279 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Minimal cover
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FORMULA
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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. Here m=2.
From Robert Israel, Feb 18 2016: (Start)
T(n,k) = C(n,k) * (2^(k-1)-1).
G.f. of triangle: x^2*y^2/((1-x)*(1-x-x*y)*(1-x-2*x*y)). (End)
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EXAMPLE
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There are 90=10+30+35+15 minimal 2-covers of a labeled 5-set.
Triangle starts:
1;
3, 3;
6, 12, 7;
10, 30, 35, 15;
15, 60, 105, 90, 31;
...
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MAPLE
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seq(seq(binomial(n, k)*(2^(k-1)-1), k=2..n), n=2..13); # Robert Israel, Feb 18 2016
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MATHEMATICA
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Table[ Binomial[n, k] (2^(k-1)-1), {n, 2, 13}, {k, 2, n}] // Flatten (* Jean-François Alcover, Sep 18 2018, from Maple *)
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PROG
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(PARI) T(n, k) = m=2; binomial(n, k)*stirling(k, m, 2)*(2^m-m-1)^(n-k); \\ Michel Marcus, Feb 18 2016
(Magma) /* As triangle: */ [[Binomial(n, k)*(2^(k-1)-1): k in [2..n]]: n in [1.. 15]]; // Vincenzo Librandi, Feb 19 2016
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CROSSREFS
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Cf. A022166, A035347, A057669, A057964, A057965, A057966, A057967, A057968.
Sequence in context: A025250 A326498 A094305 * A250301 A261954 A112434
Adjacent sequences: A057960 A057961 A057962 * A057964 A057965 A057966
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Vladeta Jovovic, Oct 17 2000
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STATUS
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approved
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