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A057963 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).

%I A057963 #34 Feb 16 2025 08:32:43
%S A057963 1,3,3,6,12,7,10,30,35,15,15,60,105,90,31,21,105,245,315,217,63,28,
%T A057963 168,490,840,868,504,127,36,252,882,1890,2604,2268,1143,255,45,360,
%U A057963 1470,3780,6510,7560,5715,2550,511,55,495,2310,6930,14322,20790,20955,14025
%N A057963 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).
%C A057963 Row sums give A000392.
%H A057963 Robert Israel, <a href="/A057963/b057963.txt">Table of n, a(n) for n = 2..10012</a> (rows 2 to 142, flattened)
%H A057963 Octavio A. Agustín-Aquino, <a href="http://arxiv.org/abs/1602.05279">Archimedes' quadrature of the parabola and minimal covers</a>, arXiv:1602.05279 [math.CO], 2016.
%H A057963 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalCover.html">Minimal cover</a>
%F A057963 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.
%F A057963 From _Robert Israel_, Feb 18 2016: (Start)
%F A057963 T(n,k) = C(n,k) * (2^(k-1)-1).
%F A057963 G.f. of triangle: x^2*y^2/((1-x)*(1-x-x*y)*(1-x-2*x*y)). (End)
%e A057963 There are 90=10+30+35+15 minimal 2-covers of a labeled 5-set.
%e A057963 Triangle starts:
%e A057963 1;
%e A057963 3, 3;
%e A057963 6, 12, 7;
%e A057963 10, 30, 35, 15;
%e A057963 15, 60, 105, 90, 31;
%e A057963 ...
%p A057963 seq(seq(binomial(n,k)*(2^(k-1)-1),k=2..n), n=2..13); # _Robert Israel_, Feb 18 2016
%t A057963 Table[ Binomial[n, k] (2^(k-1)-1), {n, 2, 13}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Sep 18 2018, from Maple *)
%o A057963 (PARI) T(n,k) = m=2; binomial(n, k)*stirling(k, m, 2)*(2^m-m-1)^(n-k); \\ _Michel Marcus_, Feb 18 2016
%o A057963 (Magma) /* As triangle: */ [[Binomial(n, k)*(2^(k-1)-1): k in [2..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Feb 19 2016
%Y A057963 Cf. A022166, A035347, A057669, A057964, A057965, A057966, A057967, A057968.
%K A057963 easy,nonn,tabl
%O A057963 2,2
%A A057963 _Vladeta Jovovic_, Oct 17 2000