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A381683 Triangle read by rows: T(n,k) = number of collections of up to k subsets of [n] covering [n], with [0]={}; n>=0, k=0..2^n.

%I A381683 #14 Mar 07 2025 06:38:11
%S A381683 1,2,0,1,2,0,1,5,9,10,0,1,14,58,125,181,209,217,218,0,1,41,401,1947,
%T A381683 6091,13987,25395,38261,49701,57709,62077,63897,64457,64577,64593,
%U A381683 64594,0,1,122,2802,30352,210448,1076880,4385616,14839576,42831176,107303376,236306016,462089756,809460556,1280895556,1846618196,2447698581
%N A381683 Triangle read by rows: T(n,k) = number of collections of up to k subsets of [n] covering [n], with [0]={}; n>=0, k=0..2^n.
%C A381683 Partial row sums of A163353.
%C A381683 For covers (collections without an empty set) see A369950.
%C A381683 For disjoint collections see A381682.
%C A381683 For disjoint covers see A102661.
%F A381683 T(n,k) = Sum_{j=0..k} Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*binomial(2^i,j).
%e A381683 Triangle begins:
%e A381683   1 2
%e A381683   0 1  2
%e A381683   0 1  5   9   10
%e A381683   0 1 14  58  125  181   209   217   218
%e A381683   0 1 41 401 1947 6091 13987 25395 38261 49701 57709 62077 63897 64457 64577 64593 64594
%e A381683   ...
%e A381683 T(3,2)=14 is the number of covering collections of 1 or 2 subsets of [3]:
%e A381683   {{1,2,3}}
%e A381683   {{},{1,2,3}}
%e A381683   {{1},{2,3}}
%e A381683   {{1},{1,2,3}}
%e A381683   {{2},{1,3}}
%e A381683   {{2},{1,2,3}}
%e A381683   {{3},{1,2}}
%e A381683   {{3},{1,2,3}}
%e A381683   {{1,2},{1,3}}
%e A381683   {{1,2},{2,3}}
%e A381683   {{1,3},{2,3}}
%e A381683   {{1,2},{1,2,3}}
%e A381683   {{1,3},{1,2,3}}
%e A381683   {{2,3},{1,2,3}}.
%t A381683 Table[Sum[Sum[(-1)^(n-i)*Binomial[n, i]*Binomial[2^i, j], {i, 0, n}], {j, 0, k}], {n, 0, 4}, {k, 0, 2^n}]//Flatten
%o A381683 (PARI) T(n,k) = sum(j=0,k, sum(i=0,n, (-1)^(n-i)*binomial(n,i)*binomial(2^i,j)));
%o A381683 for(n=0,5,for(k=0,2^n,print1(T(n,k),", "))); \\ _Joerg Arndt_, Mar 04 2025
%Y A381683 Cf. A000371 (diagonal).
%Y A381683 Cf. A102661, A163353, A369950, A381682.
%K A381683 nonn,tabf
%O A381683 0,2
%A A381683 _Manfred Boergens_, Mar 04 2025