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A095421 Triangle read by rows: T(n,m) = number of m-block proper covers (without empty blocks and without multiple blocks) of a labeled n-set (n>=2, 2<=m<=2^n-2).

%I A095421 #12 Feb 16 2025 08:32:53
%S A095421 1,6,17,15,6,1,25,230,861,1918,2975,3428,3003,2002,1001,364,91,14,1,
%T A095421 90,2125,20930,127701,568820,2003635,5820750,14282125,30030000,
%U A095421 54620475,86490950,119759325,145422600,155117515,145422675,119759850,86493225
%N A095421 Triangle read by rows: T(n,m) = number of m-block proper covers (without empty blocks and without multiple blocks) of a labeled n-set (n>=2, 2<=m<=2^n-2).
%H A095421 G. C. Greubel, <a href="/A095421/b095421.txt">Table of n, a(n) for the first 10 rows, flattened</a>
%H A095421 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ProperCover.html">Proper Cover.</a>
%F A095421 T(n, m) = Sum((-1)^(n-i)*binomial(n, i)*binomial(2^i-1, m), i=1..n) - binomial(2^n-2, m-1).
%e A095421 1;
%e A095421 6,17,15,6,1;
%e A095421 25,230,861,1918,2975,3428,3003,2002,1001,364,91,14,1;
%e A095421 ...
%t A095421 T[n_, m_] := Sum[(-1)^(n - i)*Binomial[n, i]*Binomial[2^i - 1, m], {i, 1, n}] - Binomial[2^n - 2, m - 1]; Table[T[n, m], {n, 2, 10}, {m, 2, 2^n - 2}] // Flatten (* _G. C. Greubel_, Oct 07 2017 *)
%o A095421 (PARI) for(n=2,6, for(m=2, 2^n -2, print1(sum(j=1,n, (-1)^(n-j)* binomial(n, j)*binomial(2^j -1, m)), ", "))) \\ _G. C. Greubel_, Oct 07 2017
%Y A095421 Cf. A007537(row sums), A055154, A055127, A055152, A095422, A095423.
%K A095421 easy,nonn,tabf
%O A095421 2,2
%A A095421 Goran Kilibarda, _Vladeta Jovovic_, Jun 04 2004